Wednesday, August 04, 2021

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



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

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


6. Which situation is an example of bivariate data?

1) the number of pizzas Tanya eats during her years in high school
2) the number of times Ezra puts air in his bicycle tires during the summer
3) the number of home runs Elias hits per game and the number of hours he practices baseball
4) the number of hours Nellie studies for her mathematics tests during the first half of the school year

Answer: 3) the number of home runs Elias hits per game and the number of hours he practices baseball


Bivariate means that there are two pieces of data being tracked and compared. In Choice (3), there is data for Elias scoring hits and for the hours he practices. These can be compared.

In Choice (1), only the number of pizzas is being counted.

In Choice (2), just the number of times air in put into tires is tracked.

In Choice (4), the hours studying for tests is measured, but not, say, the scores on those tests.





7. Brianna’s score on a national math assessment exceeded the scores of 95,000 of the 125,000 students who took the assessment. What was her percentile rank?

1) 6
2) 24
3) 31
4) 76

Answer: 4) 76


Given the choices provided, you don't have to do much math at all. If you beat 95,000 of 125,000, you are well above the halfway point. I could do the math to see if that would be 76 percent, but the other three numbers are so low, why bother. Choices (1), (2) and (3) are just silly.

Anyway, 95000 / 125000 = 0.76, which is the 76th percentile.





8. 8 If A = {0, 1, 3, 4, 6, 7}, B = {0, 2, 3, 5, 6}, and C = {0, 1, 4, 6, 7}, then A ∩ B ∩ C is

1) {0, 1, 2, 3, 4, 5, 6, 7}
2) {0, 3, 6}
3) {0, 6}
4) {0}

Answer: 3) {0, 6}


The intersection of two sets in the set of all elements that are present in both the first and the second set. The intersection of three sets is the set of all elements that are present in every one of the three individual sets. You can do one intersection at a time to make it less confusing, if you want to.

The intersection A ∩ B = {0, 1, 3, 4, 6, 7} ∩ {0, 2, 3, 5, 6} = {0, 3, 6}

The intersection A ∩ B ∩ C = {0, 3, 6} ∩ {0, 1, 4, 6, 7} = {0, 6}





9.Which graph represents a function?


Answer: 1)


For a graph to be a function, it must pass the vertical-line test, which states that no vertical line can be drawn through any part of the graph so that it touches more than one point on the graph.

In Choice (2), a vertical line can be drawn between, for example, (2, 2) and (2, -2). The same x-value (the input) has two different y-values (output). This is not allowed in a function.

Similarly, Choices (3) and (4) fail the test.





10. What is the product of (3x + 2) and (x - 7)?

1) 3x2 - 14
2) 3x2 - 5x - 14
3) 3x2 - 19x - 14
4) 3x2 - 23x - 14

Answer: 3) 3x2 - 19x - 14


If the four choices, the first and last terms are the same. You only need to concern yourself with the middle term.

If you use the Area Model, you can see that the four products that you need to find are:

(3x)(x) = 3x2

(3x)(-7) = -21x

(2)(x) = 2x

(2)(-7) = -14

When you combine the like terms, you see that -21x + 2x = -19x, which is Choice (3).




More to come. Comments and questions welcome.

More Regents problems.

Tuesday, August 03, 2021

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



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

More Regents problems.

June 2012

Part I: Each correct answer will receive 2 credits.


1. In a baseball game, the ball traveled 350.7 feet in 4.2 seconds. What was the average speed of the ball, in feet per second?
1) 83.5
2) 177.5
3) 354.9
4) 1,472.9

Answer: 1) 83.5


To find the unit rate of speed in feet per second divide the number of feet by the amount of seconds:

350.7 / 4.2 = 83.5

Common sense should tell you that if something travels a little over 350 feet in a little over 4 seconds, then in just ONE second, it should go about 1/4 of the distance. Choices (3) and (4) are just ridiculous. Choice (2) isn't much better, but at least it's smaller.





2. A survey is being conducted to determine if a cable company should add another sports channel to their schedule. Which random survey would be the least biased?
1) surveying 30 men at a gym
2) surveying 45 people at a mall
3) surveying 50 fans at a football game
4) surveying 20 members of a high school soccer team

Answer: 2) surveying 45 people at a mall


Bias means that you are more likely to find people who are predeisposed to give one particular answer over another. It also comes from focusing too much on one particular subgroup of people.

In Choice (1), the survey would only be asking men of their opinion, and in particular, men who are most likely athletic.

In Choice (3), the survey is specifically asking fans if they want more sports while they are attending a sporting event.

In Choice (4), the survey is again asking people involved with sports, but this time they are all of high school age, and there aren't even that many of them.

The only problem with Choice (2) is that 45 people may not seem like very many, but you will find a more random collection of people to ask.





3. The quotient of (8x5 - 2x4 + 4x3 - 6x2) / (2x2) is
1) 16x7 - 4x4 + 8x5 - 12x4
2) 4x7 - x6 + 2x5 - 3x5
3) 4x3 - x2 + 2x - 3x
4) 4x3 - x2 + 2x - 3

Answer: 4) 4x3 - x2 + 2x - 3


Divide each coefficient by 2 and reduce each exponent by 2. An exponent of 1 can be left out because it means one factor of x, which is just x. An exponent of 0 means that there are NO factors of the variable, so leave the variable off.

If should be obvious that Choice (1) is silly. It's the only one that even looks like that. Despite the word "quotient" and the fraction in the problem, Choice (1) shows the product of the two expressions.

Choice (2) is similarly wrong because it divided the coefficients but it added the expoents, instead of reducing them.

Choice (3) is incorrect because it ends with "2x - 3x", which you should wonder, why weren't those like terms combined? Because they weren't supposed to be like terms. If the original exponents were 5, 4, 3, 2, and each is reduced by 2, then you should have 3, 2, 1, 0 in your answer. But -3x has an exponent of 1.

Choice (4) is correct because -3x0 is -3.





4. Marcy determined that her father’s age is four less than three times her age. If x represents Marcy’s age, which expression represents her father’s age?
1) 3x - 4
2) 3(x - 4)
3) 4x - 3
4) 4 - 3x

Answer: 1) 3x - 4


Three times her age, x, is 3x. Four less than that means take four away from that or subtract 4 from that amount. That is 3x - 4.

4 - 3x is "obviously" wrong because her father would be less than 4 years old. In fact, if Marcy was over the age of 1, her father's age would be a negative number.





5. A set of data is graphed on the scatter plot below.
This scatter plots shows



1) no correlation
2) positive correlation
3) negative correlation
4) undefined correlation

Answer: 2) positive correlation


As the x values get bigger, the y values generally get bigger. If you were to draw a line of best fit through the data, it would have a positive slope.




More to come. Comments and questions welcome.

More Regents problems.

Monday, August 02, 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 IV: Each correct answer will receive 4 credits. Partial credit is possible.


37. A company is running a contest and offering a first, second, and third prize. First prize is a choice of a car or $15,000 cash. Second prize is a choice of a motorbike, a trip to New York City, or $2,000 cash. Third prize is a choice of a television or $500 cash.

If each prize is equally likely to be selected, list the sample space or draw a tree diagram of all possible different outcomes of first, second, and third prizes.

Determine the number of ways that all three prizes selected could be cash.

Determine the number of ways that none of the three prizes selected could be cash.

Answer:


The sample space would be (you can abbreviate, but be clear about it):

Car, motorbike, TV
Car, motorbike, $500
Car, NYC, TV
Car, NYC, $500
Car, $2000, TV
Car, $2000, $500
$15K, motorbike, TV
$15K, motorbike, $500
$15K, NYC, TV
$15K, NYC, $500
$15K, $2000, TV
$15K, $2000, $500

The tree diagram would look like this:

The tree diagram or sample space is considered the WORK for the next two questions. If you answer the questions based on what you drew, you can get credit even if you made a mistake in your diagram, as long as it is consistent.

If you did NOT create a tree diagram or sample space, then work is expected to be shown on the bottom two questions, such as using the Counting Principle.

There is only ONE way that all three prizes will be cash. ($15,000, $2,000, $500)

There are TWO ways that none of the three prizes will be cash. (Car, bike, TV, and Car, NYC, TV)





38. In right triangle ABC shown below, AC = 29 inches, AB = 17 inches, and m∠ABC = 90. Find the number of degrees in the measure of angle BAC, to the nearest degree.

Find the length of BC to the nearest inch.

Answer:


You are given that it is a right triangle, along with the length of the hypotenuse and one of the sides. You want to find the size of the angle ADJACENT to the side you know. That means using Cos.

Cos x = 17/29
x = cos-1(17/29) = 54.11...

The measure of ∠BAC is 54 degrees.

To find the length of BC, you have a choice of using the Sine ratio, or Pythagorean Theorem.

172 + BC2 = 292
289 + BC2 = 841
BC2 = 552
BC = 23.49...
BC = 23 inches

Or...

sin 54.11 = BC/29
BC = 29 * sin 54.11 = 23.49...
BC = 23 inches





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

y + x ≥ 3
5x - 2y > 10

State the coordinates of one point that satisfies y + x ≥ 3, but does not satisfy 5x - 2y > 10.

Answer:


Rewrite the inequalities in slope-intercept form, so you can use your calculator, or just plot from the y-intercept and the slope.

y + x ≥ 3
y ≥ -x + 3

5x - 2y > 10
- 2y > -5x + 10
y < 5/2 x - 5

Remember to shade on the correct sides. Remember that the second inequality is a broken or dashed line.

You may choose any point that is in the top quarter of the graph, the top of the "X", for the final answer. You may choose a point on the top part of the red line, the top part of the blue line, or any point in between them. For example, (2, 2).




End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

Sunday, August 01, 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 III: Each correct answer will receive 3 credits. Partial credit is possible.


34. The following cumulative frequency histogram shows the distances swimmers completed in a recent swim test.

Based on the cumulative frequency histogram, determine the number of swimmers who swam between 200 and 249 yards.

Determine the number of swimmers who swam between 150 and 199 yards.

Determine the number of swimmers who took the swim test.

Answer:


There are three questions and each is worth a credit.

For the first question, to find the number of swimmers who swam between 200 and 249 yards, look at the columns labeled "50–249" and "50–299". The difference between those two amounts is what you are looking for. Subtract 20 - 15 = 5. There were 5 swimmers in that range.

Similarly, to find the number between 150 and 199, look at " 50–149" and "50–199". They are both the same, so 12 - 12 = 0. There were 0 swimmers in that range.

The number of swimmers is 20 because the cumulative frequency chart counts everyone, and everyone will be included in the final column.





35. Ashley measured the dimensions of a rectangular prism to be 6 cm by 10 cm by 1.5 cm. The actual dimensions are 5.9 cm by 10.3 cm by 1.7 cm. Determine the relative error, to the nearest thousandth, in calculating the volume of the prism.

Answer:


Reminder: Relative error is a decimal. Percent of error is a percentage. Should that make a difference? Honestly, I don't think so, but the people who write the rubrics would disagree with me. And their voice is the one that matters, unfortuately.

What this means is, if you do all the work correctly and then express the answer as a correct percentage, you will lose a point, which amounts to 1/3 of the credit for the problem. Or about 33.3%.

To find the relative error, calculate the volume using Ashley's measurements. Then calculate the actual volume with the actual measurements. Find the difference between those two amounts and divide it by the actual volume. Round it to three decimal places.

Ashley's volume: V = L * W * H = 6 * 10 * 1.5 = 90

Actual volume: V = L * W * H = 5.9 * 10.3 * 1.7 = 103.309

Error: (103.309 - 90) / (103.309) = 0.1288...

To the nearest thousandth, the relative error is 0.129.





36. Solve the following system of equations algebraically for all values of x and y.

y = x2 + 2x - 8
y = 2x + 1

Answer:


Note that you could graph it or use a table of values, but in doing so, you would only get one point for a correct answer for not solving it algebraically. You can also do this first if you want to know what the answer is before you start solving it algebraically. (There's nothing wrong with doing this.)

To solve the system of equations, set the two equations equal to each other. Since each expression is equation to y, then they are equal to each other. Once you do that, you can solve the quadratic equation that you get.

y = x2 + 2x - 8
y = 2x + 1
x2 + 2x - 8 = 2x + 1
x2 - 9 = 0
(x + 3)(x - 3) = 0
x + 3 = 0 or x - 3 = 0
x = -3 or x = 3

Now that you know the values of x, go back and find the matching values of y.

y = 2(-3) + 1 = -6 + 1 = -5. (-3, -5)

y = 2(3) + 1 = 6 + 1 = 7. (3, 7)




End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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.
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 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.