Friday, July 30, 2021

(x, why?) Mini: Nonlinear Partial Differential Equations

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

What it says is what you get.

Dedicated to Dr. Anna Kiesenhofer, a mathematician researching non-linear partial differential equations, and who won an Olympic gold medal in the Women's Individual Road Race in Cycling.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
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If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


31. State the value of the expression (4.1 X 102)(2.4 X 103) / (1.5 X 107) in scientific notation.

Answer:


Remember the rules for exponents. Having base 10 is no different from having base x.

You can regroup them if you want to. Use your calculator to solve the first part. Remember to put the final answer back into scientific notation with the coefficient between 1 and 9.9999...

(4.1 X 102)(2.4 X 103) / (1.5 X 107)
(4.1)(2.4)/(1.5) X (102)(103) / (107)
6.56 X 102 + 3 - 7
6.56 X 10-2

I'm a little surprised that they didn't form the problem so that the coefficient was 10 or greater or less than one.





32. Express the product of (x + 2)/2 and (4x + 20)/(x2 + 6x + 8) in simplest form.

Answer:


With problems like this, factor whatever you can, and then cancel out common factors in the numerator and denominator which form a multiplicative identity.
(x + 2) / 2 * (4x + 20) / (x2 + 6x + 8)
(x + 2) / 2 * ( (4)(x + 5) ) / ( (x + 4)(x + 2) )
1 / 1 * ( (2)(x + 5) ) / ( (x + 4)(1) )
( (2)(x + 5) ) / (x + 4)

(2)(x + 5)
x + 4

Either (2)(x + 5) or 2x + 10 were acceptable in the numerator.





33. On the set of axes below, graph y = 3x over the interval -1 < x < 2.

Answer:


It is very important that you ONLY graph the domain that they gave you. Do NOT go beyond it. Do NOT draw arrows indicating the directions the line is going in. You CAN pick whatever scale you want for the x-axis. If you wanted, say, to count two boxes for every 1, that's fine.

Your graph should look like this:

At x = -1, the point should be at about 1/3 of the distance between 0 and 1. Do the best you can. Do NOT make a point on the x-axis.




More to come. Comments and questions welcome.

More Regents problems.

Thursday, July 29, 2021

New Improbolympic Events

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

It can add a new dimension to the games ... or take one away.

I don't think I could've done better with those hurdles, even with infinite time.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


26. What is the solution of 2 / (x + 1) = (x + 1) / 2
1) -1 and -3
2) -1 and 3
3) 1 and - 3
4) 1 and 3

Answer: 3) 1 and - 3


Cross-multiply and solve the resulting quadratic equation.

2 / (x + 1) = (x + 1) / 2
(x + 1)(x + 1) = 4
x2 + 2x + 1 = 4
x2 + 2x - 3 = 0
(x - 1)(x + 3) = 0
x - 1 = 0 or x + 3 = 0
x = 1 or x = -3





27. The total score in a football game was 72 points. The winning team scored 12 points more than the losing team. How many points did the winning team score?

1) 30
2) 42
3) 54
4) 60

Answer: 2) 42


As much as I'd like to say that 42 is the Ultimate Answer to the Question of Life, the Universe and Everything, we still have to solve this problem to see if it's the solution to this problem.

If x is the number of points the losing team scored, then x + 12 is the number of points the winning team scored because they scored 12 more points. The sum of those two expressions is x + x + 12.

x + x + 12 = 72
2x + 12 = 72
2x = 60
x = 30

The losing team scored 30. The winning team scored 30 + 12 = 42.





28. What is the perimeter of the figure shown below, which consists of an isosceles trapezoid and a semicircle?



1) 20 + 3π
2) 20 + 6π
3) 26 + 3π
4) 26 + 6π

Answer: 1) 20 + 3π


Add the three sides of the trapezoid that are shown. Do not add the dotted line because it is not part of the perimeter. It is inside. Then calculate the circumference of the semicircle.

The trapezoid sides are 4 + 6 + 10 = 20. Eliminate Choices (3) and (4).

The semicircle has a diameter of 6 because the trapezoid is isosceles. The Circumference of a circle is πd or 2πr, so the circumference of a semicircle (1/2 of a circle) is half as much: 1/2 πd or πr.

We know the diameter is 6, so the circumference is 3π. Choice (1).





29. The probability that it will rain tomorrow is 1/2. The probability that our team will win tomorrow’s basketball game is 3/5. Which expression represents the probability that it will rain and that our team will not win the game?

1) 1/2 + 3/5
2) 1/2 + 2/5
3) 1/2 * 3/5
4) 1/2 * 2/5

Answer: 4) 1/2 * 2/5


The most important word here is not. You were given the probability that the team will win. We need the probability that they will NOT win. Since there are only two possiblities, winning and not winning, and because those two probabilities must add up to 1, the probability of not winning is 1 - 3/5, which is 2/5.

The probability of two independent events occurring is the product of the two individual events happening separately.





30. The formula for the volume of a pyramid is V = 1/3 Bh. What is h expressed in terms of B and V?
1) h = 1/3 VB
2) h = V / (3B)
3) h = (3V)/B
4) h = 3VB

Answer: 3) -2 or 2


Use inverse operations to isolate the h.

V = 1/3 Bh
3V = Bh
3V/B = h




End of Part I.

More to come. Comments and questions welcome.

More Regents problems.

Wednesday, July 28, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


21. Given:

A = {1, 3, 5, 7, 9}
B = {2, 4, 6, 8, 10}
C = {2, 3, 5, 7}
D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Which statement is false?

1) A ∪ B ∪ C = D
2) A ∩ B ∩ C = { }
3) A ∪ C = {1, 2, 3, 5, 7}
4) A ∩ C = {3, 5, 7}

Answer: 3) A ∪ C = {1, 2, 3, 5, 7}


1) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, which is set D.
(1) is True.

2) A ∩ B ∩ C = {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8, 10} ∩ {2, 3, 5, 7}
= { } ∩ {2, 3, 5, 7} = { }.
(2) is True.

3) A ∪ C = {1, 3, 5, 7, 9} ∪ {2, 3, 5, 7}
= {1, 2, 3, 5, 7, 9} =/= {1, 2, 3, 5, 7, 9}.
(3) is False.

4) A ∩ C = {1, 3, 5, 7, 9} ∩ {2, 3, 5, 7} = {3, 5, 7}.
(4) is True.





22. Which expression is equivalent to ( 2x6 - 18x4 + 2x2 ) / 2x2

1) x3 - 9x2
2) x4 - 9x2
3) x3 - 9x2 + 1
4) x4 - 9x2 + 1

Answer: 4) x4 - 9x2 + 1


Divide the coefficient of each term in the numerator by 2, and reduce the exponent of the variable by 2. The rule for dividing variables with exponents is to keep the base and subtract the exponent. In this case, each term will have two fewer factors of x.

2x6 / 2x2 = x4. Eliminate Choices (1) and (3).

18x4 / 2x2 = 9x2.

2x2 / 2x2 = 1. Eliminate Choice (2).





23. In a given linear equation, the value of the independent variable decreases at a constant rate while the value of the dependent variable increases at a constant rate. The slope of this line is

1) positve
2) negative
3) zero
4) undefined

Answer: 2) negative


If one variable is increasing while the other is decreasing, the slope is negative.

If the slope were positive, both variables would be increasing or both variables would be decreasing.

A zero slope would require that the dependent variable NOT change at all but instead remain constant. For example, y = 3.

An undefined slope would require that the independent variable NOT change at all but instead remain constant. For example, x = 2.





24. The volume of a cylindrical can is 32π cubic inches. If the height of the can is 2 inches, what is its radius, in inches?

1) 8
2) 2
3) 16
4) 4

Answer: 4) 4


The Volume of a cylinder uses the formula V = π r2 h.
V = π r2 h = 32π
π r2 (2) = 32π
r2 = 16
r = 4





25. The expression ( 14 + x ) / ( x2 - 4 ) is undefined when x is
1) -14, only
2) 2, only
3) -2 or 2
4) -14, -2, or 2

Answer: 3) -2 or 2


A fraction is undefined if its denominator is equal to zero. You cannot divide by 0.

The numerator can be zero, and you can divide 0 as many ways as you want. The quotient will always be 0, unless you are divide by zero.

For what values of x is the denominator equal to 0?

x2 - 4 = 0
(x - 2)(x + 2) = 0
x - 2 = 0 or x + 2 = 0
x = 2 or x = -2




More to come. Comments and questions welcome.

More Regents problems.

Tuesday, July 27, 2021

Math!

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Math * (Math - 1) * (Math - 2) * ... * 1

It depends on the value of Math. And some people give Math more value than others!



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


16. The rectangular prism shown below has a length of 3.0 cm, a width of 2.2 cm, and a height of 7.5 cm.

What is the surface area, in square centimeters?

1) 45.6
2) 49.5
3) 78.0
4) 91.2

Answer: 4) 91.2


The surface area is the sum of the areas of the six sides of the prism. Since it is a rectangular prism, the opposite sides have the same area.

You have find the L * W, L * H, and W * H, double each of those, and find the sum.

L * W = 3 * 2.2 = 6.6

L * H = 3 * 7.5 = 22.5

W * H = 2.2 * 7.5 = 16.5

2 * (6.6 + 22.5 + 16.5) = 91.2





17. Which set of coordinates is a solution of the equation 2x - y = 11?

1) (-6, -1)
2) (-1, 9)
3) (0, 11)
4) (2, -7)

Answer: 4) (2, -7)


You can try each of the choices to see which makes a true statement, or you can rewrite the equation, solving for y, and put that into your graphing calculator and look at the Table of Values for the correct entry.

2x - y = 11
2x - 11 = y, or y = 2x - 11

Ironically, it wasn't until I typed the above equation just now that I noticed that the first three choices are all solutions to y = 2x + 11. I would've just did the mental math.





18. The graph of a parabola is represented by the equation y = ax2 where a is a positive integer. If a is multiplied by 2, the new parabola will become

1) narrower and open downward
2) narrower and open upward
3) wider and open downward
4) wider and open upward

Answer: 2) narrower and open upward


Since a is positive, it opens upward. Eliminate Choices (1) and (3).

If you double a, then the y values will increase more rapidly. That will cause the parabola to become thinner, or narrower.





19. Which equation represents a line that has a slope of 3/4 and passes through the point (2,1)?

1) 3y = 4x - 5
2) 3y = 4x + 2
3) 4y = 3x - 2
4) 4y = 3x + 5

Answer: 3) 4y = 3x - 2


The choices given are not in any familar form. However, they could easily be put into slope-intercept form by dividing by the coefficient of y to get y = mx + b.

If you divide Choices (1) and (2) by 3, then the value of m will be 4/3 instead of 3/4. Eliminate choices (1) and (2).

Next, substitute x = 2 into Choices (3) and (4).

4y = 3(2) - 2 = 6 - 2 = 4; 4y = 4, y = 1. This is the solution we are looking for.

4y = 3(2) + 5 = 6 + 5 = 11; 4y = 11, y = 11/4. This is NOT the solution we are looking for.





20. What is the value of | ( 4(-6) + 18 ) / 4! | ?
1) 1/4
2) - 1/4
3) 12
4) -12

Answer: 1) 1/4


I hate questions like this. I had a student come up to me after the exam and ask me, "Mr. Burke, what does an exclamation point in a math problem mean?" That's not to say that he shouldn't have known it, but the notation wasn't used as much as the concept it stands for. However, the reason I hate this problem is because it's meaningless and bizarre. There is no reason to put a factorial symbol in an equation like this other than to see if students were paying attention. It's a meaningless Order of Operations question with an added twist to it that only makes it longer, but not more challenging or interesting.

Order of Operations: Evaluate the numerator, multiply and add. Then evaluate the factorial, 4!, which is 4*3*2*1. Then simplify the fraction. Finally, take the absolute value.

Note that the Absolute Value is the final step, so the answer cannot be negative. Eliminate Choices (2) and (4).

| ( 4(-6) + 18 ) / 4! |
= | ( -24 + 18 ) / 4! |
= | -6 / 4! |
= | - 6/24 |
= | -1/4 |
= 1/4

I'm a little curious where they got 12 from. That should come from an "obvious" mistake, but I'm not seeing it.




More to come. Comments and questions welcome.

More Regents problems.

Monday, July 26, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


11. Is the equation A 21000(1 - 0.12)t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period?
1) exponential growth and 12%
2) exponential growth and 88%
3) exponential decay and 12%
4) exponential decay and 88%

Answer: 3) exponential decay and 12%


There is 12% decay, which will leave you with 88% of the original amount.

You know it is decay because of the minus sign, telling you that you will have less than 100% (the full amount) after the first time period.





12. The length of a rectangle is 15 and its width is w. The perimeter of the rectangle is, at most, 50. Which inequality can be used to find the longest possible width?
1) 30 + 2w < 50
2) 30 + 2w < 50
3) 30 + 2w > 50
4) 30 + 2w > 50

Answer: 2) 30 + 2w < 50


We are looking for the longest, meaning that there is a maximum number. All of the possibilities have to be LESS THAN that amount. Eliminate Choices (3) and (4).

The words "at most" 50 tell you that it can equal 50, so you want LESS THAN OR EQUAL TO.





13. Craig sees an advertisement for a car in a newspaper. Which information would not be classified as quantitative?
1) the cost of the car
2) the car's mileage
3) the model of the car
4) the weight of the car

Answer: 3) the model of the car


If you were unsure about quantitative vs qualitative, then this would be a good time to play "One of these things is NOT like the others".

Three of the four choices are numerical data and things that can be measured in dollars, miles and pounds. One of them is an arbitrary name given to the car, which is a description for the car.





14. What are the coordinates of the vertex and the equation of the axis of symmetry of the parabola shown in the graph below?


1) (0,2) and y = 2
2) (0,2) and x = 2
3) (-2, 6) and y = -2
4) (-2, 6) and x = -2

Answer: 4) (-2, 6) and x = -2


The vertex of a parabola is the turning point. It is either the maximum or the minimum point (and a parabola will only have one of those). In this example, the topmost point occurs at (-2, 6).

The axis of symmetry is a vertical line that, if drawn, would cut the parabola in half, creating two mirror images. The axis goes through the vertex.

The equation for all vertical lines is x = a, and every point on that line will be (a, y) for whatever value of y. The x coordinate of the vertex is -2.





15. A correct translation of “six less than twice the value of x” is
1) 2x < 6
2) 2x - 6
3) 6 < 2x
4) 6 - 2x

Answer: 2) 2x - 6


Do NOT confuse "less than" (subtraction) with "IS less than" (inequality).

Eliminate Choices (1) and (3) for using the < symbol instead of subtraction.

If my height is three inches less than your height, and your height is 68 inches, would you find my height by doing 68 - 3 or 3 - 68? The latter choice is ridiculous because it would give me a negative height.

"Six less than" something means "take six away from" something, and the something has to be there first to take something away from it.




More to come. Comments and questions welcome.

More Regents problems.

Sunday, July 25, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


6. Jason’s part-time job pays him $155 a week. If he has already saved $375, what is the minimum number of weeks he needs to work in order to have enough money to buy a dirt bike for $900?
1) 8
2) 9
3) 3
4) 4

Answer: 4) 4


You can set up an equation, or you can start with 375 and keeping adding 155 to it until you are over 900.

Set up an inequality in the form ax + b > c. The b represents the starting amount that he has, which is 375. This isn't repeated, so it doesn't get a variable. The a is the rate of change (the slope), which is the amount that occurs repeated, which is why it goes before the x. This is the 155.

155x + 375 > 900
155x > 525
x > 3.387...

Do NOT round down. He would have enough money after only three weeks. The first integer GREATER THAN 3.387... is 4. Jason will have money left over.





7. The expression 9a2 - 64b2 is equivalent to
1) (9a - 8b)(a + 8b)
2) (9a - 8b)(a - 8b)
3) (3a - 8b)(3a + 8b)
4) (3a - 8b)(3a - 8b)

Answer: 3) (3a - 8b)(3a + 8b)


The expression 9a2 - 64b2 is the difference of two perfect squares. Each part of it is a perfect square (which means you can take a square root of each component) and the two terms are separated by a minus sign.

When we have this, it means that the polynomial can be factored into two conjugates. Conjugates are two binomials that are identical, except that one has a plus and one has a minus separating them. When you multiply the two conjugates, you will get a zero pair with the two "middle" terms. (That is, the "OI" in "FOIL" will cancel out and become zero.)

9a2 - 64b2 = (3a - 8b)(3a + 8b). Note that the order of the two binomials doesn't matter.

Note that if you have two minus signs, then the middle terms, when combined, will not have a sum of zero.





8. The scatter plot below shows the profit, by month, for a new company for the first year of operation. Kate drew a line of best fit, as shown in the diagram.

Using this line, what is the best estimate for profit in the 18th month?
1) $35,000
2) $37,750
3) $42,500
4) $45,000

Answer: 3) $42,500


Interesting that there were two scatter plot questions so close together. (If you missed the previous post, question 4 was about correlation.)

Look at the line of best fit. Then go to 18 on the x-axis and move up to that line. You will see that the y-coordinate is between 40,000 and 45,000. So Choice 3 is the best estimate.





9. Which statement illustrates the additive identity property?
1) 6 + 0 = 6
2) -6 + 6 = 0
3) 4(6 + 3) = 4(6) + 4(3)
4) (4 + 6) + 3 = 4 + (6 + 3)

Answer: 1) 6 + 0 = 6


The identity property for addition states that adding 0 to a number doesn't change the number. It stays the same. (Multiplying by 1 for multiplication identity.)

Choice (2) shows an example of the Inverse Property of Addition, where adding two additive inverses yields a sum of zero. These two numbers are a zero pair.

Choice (3) is the Distributive Property of Multiplication over Addition.

Choice (4) is the Associative Property.





10. Peter walked 8,900 feet from home to school.

1 mile 5,280 feet

How far, to the nearest tenth of a mile, did he walk?
1) 0.5
2) 0.6
3) 1.6
4) 1.7

Answer: 4) 1.7


To change feet into miles, divide by 5,280.

8900 / 5200 = 1.71...

The sum of the leading coefficients 3 and -1 is 2, so eliminate Choice (4).

Since 8900 is greater than 5280, it's obvious that he walked more than a mile, so Choices (1) and (2) should be eliminated immediately.




More to come. Comments and questions welcome.

More Regents problems.

Friday, July 23, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


1. A system of equations is graphed on the set of axes below.


The solution of this system is
1) (0, 4)
2) (2, 4)
3) (4, 2)
4) (8, 0)

Answer: 3) (4, 2)


The solution to a system of equations is the point where the two lines intersect. That point is (4, 2). If you thought it was (2, 4), then you need to review which is the x coordinate and which is the y coordinate. Hint: the x comes first.





2. A cell phone can receive 120 messages per minute. At this rate, how many messages can the phone receive in 150 seconds?
1) 48
2) 75
3) 300
4) 18,000

Answer: 3) 300


Change minutes to seconds by dividing by 60 and then multiplying by 150 seconds.

120 / 60 * 150 = 300

If you multiplied by 60 and then divided by 150, you would've gotten 48. If you did 60 times 150 and then divided by 120 (I don't know why you would do that), you would get 75. If you just multiplied 120 by 150, you would have gotten 18,000. (It's rarely that easy when there's so much going on in a problem.)





3. The value of y in the equation 0.06y + 200 = 0.03y + 350 is
1) 500
2) 1,666.666...
3) 5,000
4) 18,333.333...

Answer: 3) 5,000


Note: On the actual exam, they used the repeater bar to indicate the repeating decimal.

0.06y + 200 = 0.03y + 350
0.03y + 200 = 350
0.03y = 150
y = 150/0.03 = 5,000

You can check your answer by substituting 5000 for y:

0.06(5000) + 200 = 0.03(5000) + 350
500 = 500 (check)





4. The scatter plot shown below represents a relationship between x and y.


This type of relationship is
1) a positive correlation
2) a negative correlation
3) a zero correlation
4) not able to be determined

Answer: 1) a positive correlation


First of all, we wouldn't say a "zero correlation". We'd say that there was no correlation. Is this splitting hairs? Maybe, but consider the difference between a zero slope and no slope. Words have meanings.

As the x values increase, the y values generally increase. So it is a positive correlation.





5. The sum of 3x2 + 5x - 6 and -x2 + 3x + 9 is
1) 2x2 + 8x - 15
2) 2x2 + 8x + 3
3) 2x4 + 8x2 + 3
4) 4x2 + 2x - 15

Answer: 2) 2x2 + 8x + 3


When you add polynomials, you combine like terms. The variables and their exponents do not change. Eliminate Choice (3).

The sum of the leading coefficients 3 and -1 is 2, so eliminate Choice (4).

5x + 3x = 8x and -6 + 9 = +3, so Choice (2) is the answer.

As the x values increase, the y values generally increase. So it is a positive correlation.




More to come. Comments and questions welcome.

More Regents problems.

Olympic Venn-ue

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Games, Sets, Matches

Not that we play with matches...



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Tuesday, July 20, 2021

Math+

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

More math content would be A+, right?

I could crack a few jokes here, or discuss possibilities. Or I could save that for future comics.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Monday, July 19, 2021

School Life #22: Getting Closer

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Sometimes you need to retreat and regroup before you face the next level.

One for the Hal-Daisy fans.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Friday, July 16, 2021

Argument of a Complex Number

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

They're not out of line here. Well, maybe above the line.

That's one problem with Algebra: there's always a lot of arguments!

The mathematical definition of Argument of a Complex Number is left as an exercise for the reader. But it's 3π/4 in this example.

It wasn't my intention to take a week off. It just worked out that way between life and the heat. Likewise, I considered doing something special for Comic #1750 (if I haven't started misnumbering again), but I settled for not doing a "mini" or a "quickie".

Final note: some of you might remember Negative Kitt from the other universe that was Through a Window, Darkly.

I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Friday, July 09, 2021

Coordinate Hustlers

(Click on the comic if you can't see the full image.)
(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Those who will not learn from Geometry will lose a lot of money to Geometry.

And as improbable as it may seem, whichever points are selected, the altitude in this problem will also be 3. The base AB is shown to be 3, so the area will always be 4.5 of those square boxes.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Thursday, July 08, 2021

June 2021 Algebra 1 Regents (v202), Part I (multiple choice)



This exam was adminstered in June 2021. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

Part II is located here.

June 2021 (v202)

Part I

Each correct answer will receive 2 credits.


1. A high school club is researching a tour package offered by the Island Kayak Company. The company charges $35 per person and $245 for the tour guide. Which function represents the total cost, C(x), of this kayak tour package for x club members?

(1) C(x) = 35x
(2) C(x) = 35x + 245
(3) C(x) = 35(x + 245)
(4) C(x) = 35 + (x + 245)

Answer: (2) C(x) = 35x + 245


$35 per person is the cost that is multiplied by the x number of people going on the tour. The $245 is paid one time before anyone has even signed up for the tour, which makes it the y-intercept.





2. The expression 3(x + 4) - (2x + 7) is equivalent to

(1) x + 5
(2) x - 10
(3) x - 3
(4) x + 11

Answer: (1) x + 5


Distribute the 3 and distribute the minus sign (-1).
3(x + 4) - (2x + 7) = 3x + 12 - 2x - 7 = x + 5



3. A function is defined as K(x) = 2x2 - 5x + 3. The value of K(-3) is

(1) 54
(2) 36
(3) 0
(4) -18

Answer: (2) 36


K(-3) = 2(-3)2 - 5(-3) + 3 = 2(9) - (-15) + 3 = 18 + 15 + 3 = 36





4. Which relation is not a function?


Answer: (4)


In a function map, each member of the domain (on the left) can only have one arrow coming out of it because each input can only can associated with one output.

In Choice (4), there are two arrows coming out of the number six, meaning that (6, 10) and (6, 12) are in the same relation. This would cause it to fail the vertical line test.



5. The value of Tony's investment was $1140 on January 1st. On this date three years later, his investment was worth $1824. The average rate of change for this investment was $19 per

(1) day
(2) month
(3) quarter
(4) year

Answer: (4) -6


1824 - 1140 = 684, which is 684 / 3 = 228 per year, which is 228 / 12 = 19 per month.

This was an interesting question in that you had to work through the choices. However, number sense should tell you that "per month" was the most likely, so you could have tried 684 / 36 first.



6. The solution to 3(x - 8) + 4x = 8x + 4 is

(1) 12
(2) 28
(3) -12
(4) -28

Answer: (4) -28


Use the Distributive Property, and then solve the equation for x.
3(x - 8) + 4x = 8x + 4
3x - 24 + 4x = 8x + 4
7x - 24 = 8x + 4
-28 = x




7. An ice cream shop sells ice cream cones, c, and milkshakes, m. Each ice cream cone costs $1.50 and each milkshake costs $2.00. Donna has $19.00 to spend on ice cream cones and milkshakes. If she must buy 5 ice cream cones, which inequality could be used to determine the maximum number of milkshakes she can buy?

(1) 1.50(5) + 2.00m > 19.00
(2) 1.50(5) + 2.00m < 19.00
(3) l.50c + 2.00(5) > 19.00
(4) l.50c + 2.00(5) < 19.00

Answer: (1) 1.50(5) + 2.00m > 19.00


If she must buy 5 ice cream cones, then substitute c = 5, not m = 5. If she can't spend more than $19, than you need to have a < symbol.



8. When written in standard form, the product of (3 + x) and (2x - 5) is

(1) 3x - 2
(2) 2x2 + x - 15
(3) 2x2 - llx - 15
(4) 6x - 15 + 2x2 - 5x

Answer: (2) 2x2 + x - 15


First of all, you know that the answer will be a quadratic expression, so Choice (1) is out. And Choice (4) is NOT is standard form, so eliminate that.

(x + 3)(2x - 5) = 2x2 + 6x - 5x - 15 = 2x2 + x - 15



9. If x = 2, y = 3 SQRT(2), and w = 2 SQRT(8), which expression results in a rational number?

(1) x + y
(2) y - w
(3) (w)(y)
(4) y / x

Answer: (3) (w)(y)


The product of SQRT(2) and SQRT(8) is SQRT((2)(8)) = SQRT(16) = 4, which is a rational number. When that result is multiplied by 3 and 2, it will still be rational. (24)



10. Which product is equivalent to 4x2 - 3x - 27?

(1) (2x + 9)(2x - 3)
(2) (2x - 9)(2x + 3)
(3) (4x + 9)(x - 3)
(4) (4x - 9)(x + 3)

Answer: (3) (4x + 9)(x - 3)


You can multiply these or graph them or factor the given expression.

If you "borrow & payback" then (4)(-27) = -108. What two factors of -108 have a sum of -3? Running through the list of factors will get you to (9)(-12). You have to "pay back" the 4.

Write out 4x2 - 12x + 9x - 27, and factor by grouping: 4x(x - 3) + 9(x - 3) = (4x + 9)(x - 3)



11. Given: f(x) = 2/3 x - 4 and g(x) = 1/4x + 1
Four statements about this system are written below.

I. f(4) = g(4)
II. When x = 12,f(x) = g(x).
III. The graphs of f(x) and g(x) intersect at (12,4).
IV. The graphs of f(x) and g(x) intersect at (4,12).
Which statement(s) are true?

(1) II, only
(2) IV, only
(3) I and IV
(4) II and III

Answer: (4) II and III


Statement I says that f(4) = 2/3(4) - 4 = -4/3 = g(4) = 1/4(4) + 1 = 2. This is not true.
Statement II says that f(12) = 2/3 (12) - 4 = 8 - 4 = 4 = g(12) = 1/4 (12) + 1 = 4. This is true.
Statement III restates statement II with extra information, which if that not only is f(12) = g(12) but they both equal 4.
Statement IV is not true because statement I is not true.



12. Which sketch represents the polynomial function f(x) = x(x + 6)(x + 3)?

Answer: Choice (1)


Choices (3) and (4) are not functions. They fail the vertical line test.

If f(x) = x(x + 6)(x + 3) then there are zeroes at x = 0, x + 6 = 0 and x + 3 = 0, or x = 0, x = -6 and x = -3.

That's Choice (1).



13. If the parent function of f(x) is p (x) = x2, then the graph of the function f(x) = (x - k)2 + 5, where k > 0, would be a shift of

(1) k units to the left and a move of 5 units up
(2) k units to the left and a move of 5 units down
(3) k units to the right and a move of 5 units up
(4) k units to the right and a move of 5 units down

Answer: (3) k units to the right and a move of 5 units up


Basic definition. You can try it in your graphing calculator by picking any value (e.g. 2) for k and observing how the graph changes.



14. Which expression is equivalent to (-4x2)3?

(1) -12x6
(2) -12x5
(3) -64x6
(4) -64x5

Answer: (3) -64x6


The exponent of 2 in (-4x2)3 goes only with the x. Everything in the parentheses gets the 3 exponent.

(-4)3 x2 * 3 = -64x6. Notice that when you take a power to another power, you have to multiply.





15. Which function has the smallest y-intercept?

Answer: Choice (1)


The y-intercept in Choice (1) is -6, as you can see from y = mx + b.

The y-intercept in Choice (2) is 1, as you can see from the row where x = 0.

The y-intercept in Choice (3) is -2, as you can see if you calculate f(0).

The y-intercept in Choice (4) is -2, as you can see on the graph at the y-axis.





16. Which domain would be the most appropriate to use for a function that compares the number of emails sent (x) to the amount of data used for a cell phone plan (y)?

(1) integers
(2) whole numbers
(3) rational numbers
(4) irrational numbers

Answer: (3) rational numbers


If you are comparing two whole numbers, you will have a ratio, and you would want rational numbers.



17. Eric deposits $500 in a bank account that pays 3.5% interest, compounded yearly. Which type of function should he use to determine how much money he will have in the account at the end of 10 years?

(1) linear
(2) quadratic
(3) absolute value
(4) exponential

Answer: (4) exponential


If it is compounded interest, then you need an exponential function. Simple interest is linear.



18. Given: the sequence 4, 7, 10, 13,. ..

When using the arithmetic sequence formula an = a1 + (n - 1)d to determine the 10th term, which variable would be replaced with the number 3?

(1) a1
(2) n
(3) an
(4) d

Answer: (4) d


The common difference, d, is 3 from term to term.



19. Below are two representations of data.
A: 2,5,5,6,6,6,7,8,9


Which statement about A and B is true?
(1) median of A > median of B
(2) range of A < range of B
(3) upper quartile of A < upper quartile of B
(4) lower quartile of A > lower quartile of B

Answer:


The medians of A and B are both 6, so eliminate Choice (1).

The range of A is 7 and the range of B is 6, so eliminate Choice (2).

The upper quartile of A is 7.5 and the upper quartile of B is 9. This is the correct statement.

The lower quartile of A and B are both 5.





20. Which system has the same solution as the system below?
x + 3y = 10
-2x - 2y = 4


(1) -x + y = 6, 2x + 6y = 20
(2) -x + y = 14, 2x + 6y = 20
(3) x + y = 6, 2x + 6y = 20
(4) x + y = 14, 2x + 6y = 20

Answer: (2) -x + y = 14, 2x + 6y = 20


This was an interesting question. Uusually when they ask something like this, the choices are multiples of the given equation (although 3 of the choices will not be multiplied correctly). This question, however, is just which two separate systems of equations happen to have the same solution, totally by coincidence.

Put these all in your graphing calulcator to find which ones intersect at the same point.

The given system has a solution of (-8, 6). Now you can graph the others, or you can plug in the values to see which gives you a true statement.

-(-8) + (6) = 14, which is =/= 6. Eliminate Choice (1).

-(8) + 6 = 14, (check), and 2(-8) + 6(6) = -16 + 36 = 20 (check). This is the solution.

If you wanted to solve the original system of equations algebraically


x + 3y = 10
-2x - 2y = 4

2x + 6y = 20
-2x - 2y = 4

4y = 24
y = 6
x + 3(6) = 10
x + 18 = 10 x = -8




21. Given the pattern below, which recursive formula represents the number of triangles in this sequence?


(1) y = 2x + 3
(2) y = 3x + 2
(3) a1 = 2, an = an - 1 + 3
(4) a1 = 3, an = an - 1 + 2

Answer: (4) a1 = 3, an = an - 1 + 2


Notice that Choices (1) and (2) are not recursive formulas, so both can be eliminated.

The first term has 3 triangles and each term after adds 2 more. This is Choice (4).



22. Students were asked to write an expression which had a leading coefficient of 3 and a constant term of -4. Which response is correct?

(1) 3 - 2x3 - 4x
(2) 7x3 - 3x5 - 4
(3) 4 - 7x + 3x3
(4) -4x2 + 3x4 - 4

Answer: (4) -4x2 + 3x4 - 4


Were my students to give me any of these answers, I'd have a problem. None of these is in standard form, which I might not care about in some instances, but if I'm asking for a specific leading coefficient, write it first!

The leading coefficient of (1) is 2. The leading coefficient of (2) is -3. The leading coefficient of (3) is 3, but the constant is 4, not -4. The leading coefficient of (4) is 3, and the constant is -4.



23. Sarah travels on her bicycle at a speed of 22.7 miles per hour. What is Sarah's approximate speed, in kilometers per minute?

(1) 0.2
(2) 0.6
(3) 36.5
(4) 36.6

Answer: (2) 0.6


Common sense should tell you that Choices (3) and (4) might be approximate answers in kilometers per hour, not minute.

To change from miles to kilometers, multiply by 1.609 (as per the back of the booklet). To change hours to minutes, divide by 60.

22.7 * 1.609 / 60 = 0.608..., which is approximately 0.6.



24. Which ordered pair does not fall on the line formed by the other three?

(1) (16,18)
(2) (12,12)
(3) (9,10)
(4) (3,6)

Answer:


Another interesting problem in that you are required to look at all for points to see with one doesn't belong.

The easiest thing to do is plot the four points on graph paper and see which one doesn't line up.

If you checked the slopes, you would have seen that lines from (16, 18) to the other three all would have different slopes.

(18 - 12) / (16 - 12) = 6/4 or 3/2, (18 - 10) / (16 - 9) = 8/7, (18 - 6) / (16 - 3) = 12 / 13. Since all three are different, you can be sure that (16, 18) is the odd point out.








More to come. Comments and questions welcome.

More Regents problems.

Wednesday, July 07, 2021

Why Did I -- ?

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Neither one was me this morning, but I was closer to the second guy.

I don't know what possesses people to stop in doorways and at the tops and bottoms of escalators, thinking that these are the best places to evaluate their surroundings.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.



Tuesday, July 06, 2021

June 2021 Algebra 1 Regents (v202), Parts III & IV



This exam was adminstered in June 2021. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

Part II is located here.

June 2021 (v202)

Part III

Each correct answer will receive 3 credits. Partial credit can be earned. It is usually possible to get 1 point for a correct answer with no correct work shown.


33. Joey recorded his heartbeats, in beats per minute (bpm), after doing different numbers of jumping jacks. His results are shown in the table below.

[TABLE COMING]
(x, y), (0, 68), (10, 84), (15, 104), (20, 100), (30, 120)

State the linear regression equation that estimates the heart rate per number of jumping jacks

State the correlation coefficient of the linear regression equation, roudned to the nearest hundredth

Explain what the correlation coefficient suggest in the context of this problem.

Answer:


Put the data into a table in your graphing calculator. Then go to Stats and perform a linear regression.

You will get the a = 1.72 and b = 69.4. Written in the form y = ax + b, that gives you y = 1.72x + 69.4

The correlation coefficient is 0.967..., which is 0.97 to the nearest hundredth.

In the context of this problem, this suggests a very strong positive correlation between the number of jumping jacks and heartbeats per minute.





34. Hannah went to the school store to buy supplies and spent $16.00. She bought four more penciels than pens and two fewer erasers than pens. Pens cost $1.25 each, pencils cost $0.55 each, and erasers are 0.75 each.

If x represents the number of pens Hannah bought, write an equation in terms of x than can be used to find how many of each item she bought.

Use your equation to determin algebraically how many pens Hannah bought.


Answer:


x = the number of pens Hannah bought
(x + 4) = the number of pencils Hannah bought
(x - 2) = the number of erasers Hannah bought

The equation to solve would be 1.25x + 0.55(x + 4) + 0.75(x - 2) = 16

1.25x + 0.55(x + 4) + 0.75(x - 2) = 16.00
1.25x + 0.55x + 4 * 0.55 + 0.75x - 2 * 0.75 = 16.00
1.25x + 0.55x + 2.20 + 0.75x - 1.50 = 16.00
1.25x + 0.55x + 0.75x = 15.30
2.55x = 15.30
x = 6

Note: if you didn't get a Whole Number, you know that you must've made a mistake. You can't buy part of a pen or pencil or eraser.

x = the number of pens Hannah bought = 6
(x + 4) = the number of pencils Hannah bought = 6 + 4 = 10
(x - 2) = the number of erasers Hannah bought = 6 - 2 = 4

You can check your work: 1.25(6) + 0.55(6 + 4) + 0.75(6 - 2) = 1.25(6) + 0.55(10) + 0.75(4) = 16 (check!)



35. Graph the given set of inequalities on the set of axes below:

y < -3/4 x + 5
3x - 2y > 4

Is (6, 3) a solution to the system of inequalities? Explain your answer.

Answer:


Remember: when answering the final question, answer it based on YOUR graph, even if you made a mistake on your graph. The graph is your justification. As long as you are consistent, you will get a point for your answer. If you did not do the graph, you cannot get a point for "Yes" or "No" unless you back it up with some work, such as substitution into both inequalaities.

The first inequality can be graphed as is because you have a slope and a y-intercept. Since it has a "<" sign, it will be a solid line with shading beneath it. All points on the line and below it are solutions to the inequality.

The second inequality needs to be rewritten into a more useful form. (Standard form can be useful, too, but in this case the x-intercept is (4/3, 0), which would be difficult to graph.)

Do the following:

3x - 2y > 4
-2y > -3x + 4
y < 3/2x - 2

The "<" symbol means it will be a broken or dashed line with shading beneath it. The line itself is a boundary and none of those points are solutions.

Put a captial "S" in the section of the graph with double-shading (the criss-cross pattern shown below). That is the section that is the solution to the system. Be sure to label the lines.

According to the graph, (6, 3) is not a solution because it is not in the double-shaded section of the graph.





36. A ball is projected up into the air from the surface of a platform to the ground below. The height of the ball above the ground, in feet, is modeled by the function f(t) = -16t2 + 96t + 112, where t is the time, in seconds, after the ball is projected.

State the height of the platform in feet.

State the coordinates of the vertex. Explain what it means in the context of this problem.

State the entire interval over which the ball's height is decreasing.


Answer:


This is a problem of projectile motion that uses gravity (the -16 leading coefficient is a giveaway!). It's a quadratic function and will follow the path of a parabola with the following limitations: the independent variable cannot be negative because that would be negative time, and the dependent variable cannot be negative because that would be below the ground. In reality, the ball will bounce when it strikes the ground, and the function does not model this.

The height of the platform is 112, which is the y-intercept. When t = 0, the ball has not left the platform yet, so the platform is 112 feet.

The cooridinates of the vertex can be found in your calculaor, or you can use the formula for the Axis of Symmetry:


x = -b / (2a)
x = -96 / (2(-16)) = -96 / -32 = 3
h(3) = -16(3)2 + 96(3) + 112 = 256

The vertex is (3, 256) which is the ball will reach its highest point of 256 feet at 3 seconds.

The ball will be decreasing after t = 3 until it hits the ground. The ground has a height = 0. We have to find the zeroes of the equation.

-16t2 + 96t + 112 = 0
t2 - 6t + 7 = 0
(t - 7)(t + 1) = 0
t = 7 or t = -1
Discard t = -1 because time cannot be negative.

The ball is decreasing over the interval 3 < t < 7.

Part IV

Each correct answer will receive 4 credits. Partial credit can be earned. It is usually possible to get 1 point for a correct answer with no correct work shown.





37. At a local shop, the price of plants includes sales tax.

The cost of 4 large plants and 8 medium plants is $40. The cost of 5 large plants and 2 medium plants is $28.

If L is the cost of a large plant and M is the cost of a medium plant, write a system of equatios that models this situation.

Could the cost of one large plant be $5.50 and the cost of one medium plants be $2.25? Justify your answer.

Determine algebraically both the cost of a lare planst and the cost of a medium plant.


Answer:


Note that the problem actually used l and m (lower case). However, on this blog it would be difficult to distinguish the l from a 1 (one). To avoid confusion, you should either use a capital L or a script l that doesn't look like a 1.

Translate the two sentences into a system of equations.
4L + 8M = 40
5L + 2M = 28

To find if (5.50, 2.25) is a solution, substitute these numbers into BOTH equations. If they do not work for both equations, then it isn't a possible solution.
4(5.50) + 8(2.25) = 40 (check!)
5(5.50) + 2(2.25) = 32, not 28. This is not a solution.

To solve the system, multiply the second equation by 4, then subtract the equations:


4L + 8M = 40
5L + 2M = 28
4L + 8M = 40
20L + 8M = 112
16L = 72
L = 4.50
4L + 8M = 40
4(4.50) + 8M = 40
18 + 8M = 40
8M = 22
M = 2.75

The cost of a Large plant is $4.50 and the cost of a Medium plant is $2.75.




End of Exam






Comments and questions welcome.

More Regents problems.

Monday, July 05, 2021

June 2021 Algebra 1 Regents (v202), Part II



This exam was adminstered in June 2021. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2021 (v202)

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Solve algebraically for y:


4(y - 3) < 4(2y + 1)

Answer:


The Distributive property is unnecessary but not incorrect if you use it. Since both sides have a multiplier of 4, the inequality can be divided by 4.

4(y - 3) < 4(2y + 1)
y - 3 < 2y + 1
-3 < y + 1
-4 < y
or y > -4
(rewriting to put the y on the left is not necessary, however, if you do, you must flip the inequality symbol)





26. Graph the function f(x) = | (1/2)x + 3 | over the invertal -8 < x < 0.

Answer:


If you are given an interval, DO NOT go beyond it, DO NOT use arrows. You should include solid (closed) circles at the two endpoints.

An absolute value graph is V-shaped. This one will have a slope on -1/2 on the left side of the vertex and +1/2 on the right side.

The vertex is (-6, 0) because the function can be written as f(x) = -1/2 |x + 6| + 0, with vertex (h, k) being (-6, 0).

You can use your graphing calculator, or you can just make a Table of Values by plugging in -8, -7, -6, -5, etc. Make sure you keep going until you hit the vertex.





27. The table below shows the height in feet, h(t), of a hot-air balloon and the number of minutes, t, the balloon is in the air.
[TABLE TO BE ADDED]

The function h(t) = 30.5t + 8.7 can be used to model this table.

Explain the measning of the slope in the context of the problem.

Explain the meaning of the y-intercept in the context of the problem.

Answer:


The table is actually unnecessary for answering the questions because you already have the function that models it.

The slope, 30.5, represents how many feet the balloon is rising each minute.

The y-intercept, 8.7. represents the ballon's height from the ground before it started rising.



28. Factor x4 - 1 completely.

Answer:


Any time you see "factor completely", you can be pretty sure that there will be more than one step. Likewise, if you see an exponent of 4 (or generally anything above 2), you should suspect that there will be an extra step.
x4 - 1
= (x2 + 1)(x2 - 1)
= (x2 + 1)(x + 1)(x - 1)

As you can see, the original expression was the Difference of Two Sqaures, which could be factored into two conjugates (such as (x+a)(x-a)). The square root of x4 is x2 because 2 is half of 4, and (x2)2 = x2 * x2 = x4.

However, x2 - 1 is also the Difference of Two Squares and can also be factored the same way. On the other hand, x2 + 1 has no real roots.





29. Mike knows that (3, 6.5) and (4, 17.55) are points on the graph of an exponential function, g(x), andhe wants to find another point on the graph of this function.

Fist, he subtracts 6.5 from 17.55 to get 11.05.

Next, he adds 11.05 and 17.55 to get 28.6.

He states that (5, 28.6) is a point on g(x).

Is he correct? Explain your reasoning.

Answer:


This is one of those times where if you say "Yes" or "No" without any explanation, you will not get any credit. You can't get a point for a 50-50 guess.

If you have the correct answer but answer but your reasoning isn't completely correct, you may only get one credit. And if you make a conceptual error (see below), you could get one credit as long as you are consistent.

No, Mike is not correct because he was trying to find slope of a straight line between the two points he had, but it is a linear function. This gives you 2 credits.

If you said, "Yes because Mike correctly calculated the slope between the two points and used that to find another point on the line", then you may have been awarded one point for the conceptual error of confusing a linear and an exponential function.





30. Use the method of completing the square to determine the vertex of f(x) = x2 - 14x - 15.

State the coordinates of the vertex.

Answer:


You can graph this to find the vertex in advance, or use the formula for the Axis of Symmetry. It may be easier to do if you already find the answer another way. You can get a point for stating the coordinates of the vertex if you find it using a different method than Completing the Square.

To Complete the square, keep in mind that the Axis of Symmetry is x = -b/(2a). You want to cut that middle coefficient in half. If you recall, (x + 1)2 = x2 + 2x + 1, and (x + 4)2 is x2 + 8x + 16. The middle term is always double and the last term must be a perfect square.

f(x) = x2 - 14x - 15
half of -14 is -7, so (x - 7) will be in our answer; (-7)2 = 49
f(x) + 49 = x2 - 14x + 49 - 15
f(x) + 49 = (x - 7)2 - 15
f(x) = (x - 7)2 - 64

The vertex is (7, -64)





31. The temperature inside a cooling unit is measured in defrees Celsius, C. Josh wants to find out how cold it is in degrees Fahrenheit, F.

Solve the formula C = 5/9 (F - 32) for F so that Josh can convert Celsius to Fahrenheit.

Answer:


Seriously, everyone should be able to do this, even outside of a math class. This is one of the most common examples given in math text books, and it's useful to know if you ever travel outside the U.S., or if you are from outside the U.S. and travel here.

C = 5/9 (F - 32)
9C = 5(F - 32)
9/5 C = F - 32
9/5 C + 32 = F
F = 9/5 C + 32





32. Solve 4w2 + 12w - 44 = 0 algebraically for w, to the nearest hundredth.

Answer:


Since they are asking for the nearest hundredth, you know that the equation isn't going to factor "nicely" and that you will have to use the quadratic formula. At least in this case, they didn't ask you to simplify the radical, which would have made it a more complicated problem.

There are two approaches you can take, and I will show them both because I am so Awesome, and modest, too.

4w2 + 12w - 44 = 0
w2 + 3w - 11 = 0

w = ( -3 + ( (-3)2 - 4(1)(-11) )(1/2) ) / (2(1))
w = ( -3 + ( 9 + 44 )(1/2) ) / 2
w = ( -3 + ( 53 )(1/2) ) / 2
w = 2.1400... or w = -5.1400...
w = 2.14 or w = -5.14




End of Part II






More to come. Comments and questions welcome.

More Regents problems.

Sunday, July 04, 2021

Happy Independence Day 2021!

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Celebrating our Independent Variables Everywhere with a Liberty Bell Curve!



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Friday, July 02, 2021

Algebra Problems of the Day (Integrated Algebra Regents, January 2013)



While I'm waiting for new Regents exams to come along (AND THEY ARE COMING SOON!), I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part IV: Each correct answer will receive 4 credits. Partial credit can be earned.


37. Using the line provided, construct a box-and-whisker plot for the 12 scores below.

26, 32, 19, 65, 57, 16, 28, 42, 40, 21, 38, 10

Determine the number of scores that lie above the 75th percentile.

Answer:


Before you do anything else, PUT THE NUMBERS IN ORDER!

There are 12 scores, so the median is between the sixth and seventh (which is the sixth from the back). That means Q1 is between the third and fourth, and Q3 is between the ninth and tenth. You don't need to "take the average", just find the number in between them. In each case, it's only 1 or 2 away. No need to "add them and divide by 2".

The minimum is the first (lowest) number, and the maximum is the last (highest) number. You should label all five of these numbers so that they are somewhere on the page. In particular, there is no obvious way from the number line to know that Q3 is 41.

Pick a scale for the number line. It is big enough to count by fives. You DO NOT have to start at 0. I did, but there isn't any particular reason to. You could have started with 5 or 10.

Do NOT label the number line with the scores. What I mean here: don't write 10, 16, 19, etc. under the tick marks. That's not how a number line works. And I've seen it written by students who didn't understand what they were supposed to do.

Since the third-quartile (Q3) is 41, there are THREE numbers greater than it. To get this credit, you need to have Q3 stated somewhere on the page, and you should identify those three numbers. You can write the three numbers, or where you wrote them in order, you can circle them and write that they are above the Q3.

The image below shows the box-and-whisker plot.





38. A metal pipe is used to hold up a 9-foot fence, as shown in the diagram below. The pipe makes an angle of 48° with the ground.

Determine, to the nearest foot, how far the bottom of the pipe is from the base of the fence.

Determine, to the nearest foot, the length of the metal pipe.

Answer:


You are given an angle and the length of the opposite side. That means that you have two trigonometric functions avaiable to you:
sin (48°) = opposite / hypotenuse
tan (48°) = opposite / adjacent

Use the sine function to find the length of the metal pipe, and the tangent function to find the distance on the ground. You will get half credit if you reverse these. You will lose points if you cosine in place of one of the other two functions.

Some common sense: the metal pipe MUST be longer than 9 feet because it's the hypotenuse. It you get a number smaller than 9, you made a mistake. Since the angle is greater than 45, that means that the adjacent side will be smaller than opposite side, but since it is 48 degrees, it will not be shorter by a lot.

Finally, make sure your calculator is in DEGREE mode not RADIAN mode or you will get crazy results. Conversely, if you get crazy results, check if your calculator is in RADIAN mode.

tan (48°) = 9 / x
x = 9 / tan (48°) = 8.1...
The bottom is approximately 8 feet from the fence.

sin (48°) = 9 / x
x = 9 / sin (48°) = 12.1...
The metal pipe is approximately 12 feet long.





39. On the set of axes below, graph the following system of equations.

y + 2x = x2 + 4
y - x = 4

Using the graph, determine and state the coordinates of all points in the solution set for the system of equations.

Answer:


You are graphing a parabola and a line. They may intersect at one point, two points or zero points. Note that even if you make graphing errors, you can get points for stating the solutions (even if you have to approximate a decimal) that are shown ON YOUR GRAPH. If your two lines don't intersect, then you must state that there are No Solutions to the system.

The first mistake that might be made usually happen when rewriting the equations to isolate the y on the left. This will allow you to use your graphing calculator to help you create a Table of Values.

y + 2x = x2 + 4
y = x2 - 2x + 4
This will have its Axis of Symmetry at x = 1.

y - x = 4
y = x + 4
Slope = 1, y-int = 4

Since the parabola is symmetrical, if you start with x = 1, 2, 3, 4, you will get the results for x = 0, -1, -2 as well.

As you can see in the image below, there are two points of intersection, (0, 4) and (3, 7). You can label these points on the graph. However, if you identify any other points on the graph, then you need to label these as your solutions.

Also, AT LEAST ONE of the two lines must be labeled with the original equation from the problem, not the rewritten one. Yes, it may be "obvious" to you which line is the quadratic and which is linear, but it must be stated. Not doing this is a graphing error and you will lose a point.




End of Part IV
End of Exam.


How'd you do?



More to come. Comments and questions welcome.

More Regents problems.

Thursday, July 01, 2021

School Life #21: School's Out 2021

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You can't just go breaking a fourth wall like that!

The more things change, the more they'll stay the same. It's a weird cosmic, temporal vortexual balancing act. With generic building blocks.



I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.





Come back often for more funny math and geeky comics.