Monday, February 19, 2024

January 2024 Algebra 2, Part II


This exam was adminstered in January 2024.

More Regents problems.

Algebra 2 January 2024

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. Factor the expression x3 + 4x2 - 9x - 36 completely.

Answer:


Factor by grouping, and then factor the quadratic you get after the first step.

There are two ways to group, and either should work in any question of this kind.

x3 + 4x2 - 9x - 36
(x3 + 4x2) - (9x + 36)
x2(x + 4) - 9(x + 4)
(x2 - 9)(x + 4)
(x + 3)(x - 3)(x + 4)

You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.

x3 - 9x + 4x2 - 36
(x3 - 9x) + (4x2 - 36)
x(x2 - 9) + 4(x2 - 9)
(x + 4)(x2 - 9)
(x + 4)(x + 3)(x - 3)

Note: This was Very Similar to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3).





26. Determine if x + 4 is a factor of 2x3 + 10x2 + 4x - 16. Explain your answer.

Answer:


If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4.

2(-4)3 + 10(-4)2 + 4(-4) - 16 = 0

Since the expression is equal to zero when x = -4, then (x + 4) must be a factor.

You could also solve this using polynomial division.

(x + 4) divides evenly, with no remainder, so it is a factor.





27. An initial investment of $1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)4t, where t is the time in years.
Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7.

Answer:


Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer).

V(7) = 1000(1.01)4(7) = 1321.29

V(2) = 1000(1.01)4(2) = 1082.86

Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is $48 to the nearest dollar.





28. When ( 1 / ∛(y2) ) y4 is written in the form yn, what is the value of n? Justify your answer.

Answer:


Use the laws of exponents to change the radical into a fraction. The combine the terms.

( 1 / ∛(y2) ) y4
( 1 / (y2/3) y4
(y-2/3) y4
y10/3

n = 10/3.





29. The heights of the members of a ski club are normally distributed. The average height is 64.7 inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall.

Answer:


They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this.

You need to use the normalcdf function.

Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%.

All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation.





30. The explicit formula an = 6 + 6n represents the number of seats in each row in a movie theater, where n represents the row number. Rewrite this formula in recursive form.

Answer:


A recursive function needs an initial value (a1) and an equation for an is terms of an-1.

The inition value a1 = 12.

Then an = an-1 + 6, because the common difference (rate of change) is 6.





31.Write (2xi3 - 3y)2) in simplest form.

Answer:


Square the binomial, substitute the powers of i, and Combine Like Terms.

(2xi3 - 3y)2)

(2xi3 - 3y)(2xi3 - 3y)

4x2i6 - 6xyi3 - 6xyi3 + 9y2

-4x2 - 12xyi3 + 9y2

-4x2 + 12xyi + 9y2





32. A survey was given to 1250 randomly selected high school students at the end of their junior year. The survey offered four post-graduation options: two-year college, four-year college, military, or work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can be made about the population of high school juniors, based on this survey

Answer:


This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%.

One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option.




End of Part II

How did you do?





More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



Friday, February 16, 2024

January 2024 Geometry Regents Part IV


This exam was adminstered in January 2024.

January 2024 Geometry, Part IV

A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.


35. Quadrilateral MATH has vertices with coordinates M(-1,7), A(3,5), T(2,-7), and H(-6,-3).
Prove that quadrilateral MATH is a trapezoid.
[The use of the set of axes on the next page is optional.]

State the coordinates of point Y such that point A is the midpoint of MY.

Prove that quadrilateral MYTH is a rectangle. [The use of the set of axes below is optional.]

Answer:


You don't neeed to use the grid but it may be helpful for visualizing. If you do use it, you still need to answer the questions fully and completely. You can't rely on whatever is in the grid to be sufficient.

To show that MATH is a trapezoid, you have to that there is one pair of parallel sides. You can do this by find the slopes of the four sides. This is easy if you graph it because you can count boxes without worrying about subtracting signed numbers (if that's a problem for you). Only two of the sides will be the same and the other two will be different.

Slope MA (7 - 5) / (-1 - 3) = 2/-4 = -1/2

Slope AT = (-7 - 5) / (2 - 3) = -12/-1 = 12

Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2

Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Since there is one pair of parallel sides, the quadrilateral is a trapezoid.

To find point Y, find the change in x-value and y-value fro M to A and add those numbers again to get Y.

M(-1,7) -> A(3,5) is a translation of +4,-2. Y(3+4,5-2) = Y(7,3)
This was worth a point even if you didn't show work.

To show that MYTH is a rectangle, you could show that the opposite sides are parallel (same slope) and that two consecutive sides are parallel (slopes are inverse reciprocals).

Slope MY (7 - 3) / (-1 - 7) = 4/-8 = -1/2

Slope YT = (-7 - 3) / (2 - 7) = -10/-5 = 2

You previously found:
Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2
Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Opposite sides are parallel, so it is a parallelogram.

(-1/2)(1) = -1. MY is perpendicular to YT, so angle Y is a right angle. Therefore, MYTH is a rectangle.




End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



January 2024 Algebra 1 Regents Part IV



This exam was adminstered in January 2024.

More Regents problems.

January 2024

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.


37. Jim had a bag of coins. The number of nickels, n, and the number of quarters, q, totaled 28 coins. The combined value of the coins was $4.

Write a system of equations that models this situation.

Use your system of equations to algebraically determine both the number of quarters, q, and the number of nickels, n, that Jim had in the bag.

Jim was given an additional $3.00 that was made up of equal numbers of nickels and quarters. How many of each coin was he given? Justify your answer.

Answer:


Write one equation for the total number of the nickels and quarters, and then write a second equation for the total value of those nickels and quarters.
n + q = 28
5n + 25q = 400

Use substition or elimination to solve the equation. For example, you could replace q with 28 - n.

5n + 25(28 - n) = 400

5n - 25n + 700 = 400

-20n = -300

n = 15

15 + q = 28

q = 13

If he was given an equal amount of nickels and quarters, then n = q. Therefore,

5q + 25q = 300
30q = 300
q = 10
n = 10

He received 10 more nickels and 10 more quarters.




End of Exam

How did you do?





More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



Thursday, February 15, 2024

January 2024 Geometry Regents Part III


This exam was adminstered in January 2024.

January 2024 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.


32. Trish is a surveyor who was asked to estimate the distance across a pond. She stands at point C, 85 meters from point D, and locates points A and B on either side of the pond such that A, D, and B are collinear.


Trish approximates the measure of angle DCB to be 35° and the measure of angle ACD to be 75°.
Determine and state the distance across the pond, AB, to the nearest meter.

Answer:


Using the 75 degree angle, you can find the length of AD, which we'll call x. Using the 35 degree angle, you can find the length of DB, which we'll call y. Add the two together to find the length of AD.

In both cases, we have the opposite side and we need to find the adjacent side. So we need to use the tangent ratio twice.

tan 75 = x / 85
x = 85 * tan 75 = 317.224...

tan 35 = y / 85
y = 85 * tan 35 = 59.517...

AB = x + y = 317.224 + 59.517 = 376.741 = 377 meters




33. A candle in the shape of a right pyramid is modeled below. Each side of the square base measures 12 centimeters. The slant height of the pyramid measures 16 centimeters.


Determine and state the volume of the candle, to the nearest cubic centimeter
The wax used to make the candle weighs 0.032 ounce per cubic centimeter. Determine and state the weight of the candle, to the nearest ounce.

Answer:


Notice that they gave you the slant height and not the height. You need the height to find the Volume. If you take a vertical slice (cross-section) of the pyramind, you would get an isosceles triangle with a base of 12 and two legs that were 16 cm. If you draw an altitude, you will get two congruent right triangles with a base of 6 and a hypotenuse of 16. Use this information to find the height.

(6)2 + b2 = (16)2
36 + b2 = 256
b2 = 220
b = √(220) = 14.832...

Use this value to find the Volume.

V = (1/3) Area of Base * height = (1/3) * 12 * 12 * 14.832 = 711.936 = 712 cu cm.

The weight is equal to the Volume times the Density: W = (712) * (0.032) = 22.784 = 23




34. In the diagram of quadrilateral ABCD below, AB ≅ CD, and AB || CD.
Segments CE and AF are drawn to diagonal BD such that BE ≅ DF.


Prove: CE ≅ AF

Answer:


TO prove that CE is congruent to AF, you are going to have to show that triangles BEC and DFA are congruent and then use CPCTC. To show that the triangles are congruent, you can use SAS.

Your proof should look like this:

Statement Reasons
Quadrilateral ABCD, AB ≅ CD, AB || CD, and BE ≅ DF. Given
ABCD is a parallelogram A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
BC ≅ AD Opposite sides of parallelograms are congruent.
BC || AD Opposite sides of parallelograms are parallel.
∠ CBE ≅ ∠ ADF Alternate Interior Angles
△BCE ≅ △DAF SAS Postulate
(AB) / (AE) = (TR) / (TE) Corresponding sides of similar triangles are proportional
CE ≅ AF CPCTC

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



Tuesday, February 13, 2024

January 2024 Algebra 1 Regents Part III


This exam was adminstered in January 2024 .

More Regents problems.

January 2024

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


33. While playing golf, Laura hit her ball from the ground. The height, in feet, of her golf ball can be modeled by h(t) = - 16t + 48t, where t is the time in seconds.
Graph h(t) on the set of axes below.


What is the maximum height, in feet, that the golf ball reaches on this hit?
How many seconds does it take the golf ball to hit the ground?

Answer:


Look at the graph below. You can put the equation in the graphing calculator. You might want to set it so that it shows every 0.5 increment of x because the high point is going to happen at x = 1.5

The maximum height the ball reaches is 36 feet (which happens at 1.5 seconds).

It takes 3 seconds for the ball to hit the ground.





34. The table below shows the number of SAT prep classes five students attended and the scores they received on the test.

State the linear regression equation for this data set, rounding all values to the nearest hundredth.
State the correlation coefficient, rounded to the nearest hundredth.
State what this correlation coefficient indicates about the linear fit of the data.

Answer:


Enter all the data into two lists in your graphing calculator. You will have to run a linear regression. Make sure you have DIAGNOSTICS ON set on your calculator.

When you run the linear regression, you will get a = 40.48 and b = 363.81, rounded to the nearest hundredth.
So the equation is y = 40.48x + 363.81

The correlation coefficient, r, is 0.84.

There is a strong positive correlation between the number of SAT prep courses attended and the score on the Math SAT.





35. Julia is 4 years older than twice Kelly’s age, x. The product of their ages is 96.
Write an equation that models this situation.
Determine Kelly’s age algebraically.
State the difference between Julia’s and Kelly’s ages, in years.

Answer:


Kelly's age is x. Write an expression for Julia in terms of x. The product of that expression and x will be 96. Solve the quadratic equation that results from it.

J = 2x + 4
x(2x + 4) = 96

2x2 + 4x = 96
2x2 + 4x - 96 = 0
x2 + 2x - 48 = 0
(x + 8)(x - 6) = 0
x + 8 = 0 or x - 6 = 0
x = -8 or x = 6

Throw out the negative answer because age cannot be negative. Therefore, Kelly is 6 years old.

Julia is 2(6) + 4 = 16. The difference between their ages is 10 years.

If you messed up the signs and thought that Kelly was 8 years old, then Julia would 20, and the difference would be 12 years. If you made one mistake, you would have lost only one point if the rest of your answers were consistent with that mistake.





36. On the set of axes below, graph the following system of inequalities:
2x - y > 4
x + 3y > 6

Label the solution set S.

Is (4,2) a solution to this system? Justify your answer.

Answer:


Rewrite the inequalities into slope-intercept form. Remember when you divide an inequality by a negative number, you have to flip the direction of the inequality symbol.

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

x + 3y > 6
3y > -x + 6
y > -1/3 x + 2

Both inequalites will have broken lines. Shade above the line y > 1/3x + 2, and below y < 2x - 4. Mark the area with the crisscross with a big "S". This is your solution.

(4,2) is a solution to the system because it's in the double-shaded area.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



Monday, February 12, 2024

A Fine Line

(Click on the comic if you can't see the full image.)
(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

There's a fine line between humor and informational context.

You know there's a great chance that Mike is not only aware that it was a joke but it's something that he would've said.

And everyone likes Mike. Right? And I ask this not because people say he's my avatar in this comic.

I almost had Scott tell it to Ken, but their rivalry is a little different (bordering, perhaps, on animosity).



I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.





Come back often for more funny math and geeky comics.



Saturday, February 10, 2024

January 2024 Geometry Regents Part II


This exam was adminstered in January 2024 .

January 2024 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.


25. In isosceles triangle ABC shown below, AB ≅ AC, and altitude AD is drawn.

The length of AD is 12 cm and the length of BC is 10 cm.
Determine and state, to the nearest cubic centimeter, the volume of the solid formed by continuously rotating triangle ABC about AD.

Answer:


Rotating triangle ABC around AD would create a cone with a height equal to AD (which is 12 cm) and a radius equal to BD (which is half of 10, or 5).

Use the formula for the Volume of a cylinder.

V = (1/3) π r2 h = (1/3) π (5)2 (12) = 314.159 ... = 314

Amusingly, 12 * 25 = 300, and 1/3 of 300 is 100, so the answer is 100 pi, rounded.

(Okay, that might only be amusing to me, but I'm amused and you may choose to be as well.




26. The diagram below models the projection of light from a lighthouse, L. The sector has a radius of 38 miles and spans 102°.


Determine and state the area of the sector, to the nearest square mile.

PLEASE, PLEASE, PLEASE!!! Make sure you round correctly. It hurts to see students do all the work to answer this question and then lose half of the points because they didn't round correctly.

Answer:


You find the area of a sector in the same way that you find the area of an entire circle. However, in the same way that a cone is only 1/3 of a cylinder of the same dimensions, a sector is only a fraction of the entire circle, and that fraction will be the central angle divided by 360 degrees.

Use the formula for the Area of a Circle and multiply by the fraction.

A = (102/360) π r2 = (102/360) π (38)2 = 1285.33... = 1285 square miles.




27. A Segment CA is drawn below. Using a compass and straightedge, construct isosceles right triangle CAT where CA is perpendicular to CT and CA ≅ CT. [Leave all construction marks.]

Answer:


Extend line CA past point C (to the left). Draw a semicircle with radius AC, mark the new point on line AC and call it point D.

Construct a perpendicular bisector of DA through point C. Put the compass and point D and open it wider than the length of DC. Make arcs above and below line DA. Move the compass to point A without changing the width of the compass. Make marks above and below DA that intersect the arcs you just make.

Draw the vertical line between the two intersections and through point C.

Label one of the intersections point T. Use a straightedge to draw CT and TA.

You have constructed isosceles triangle CAT. Done.




28. A On the set of axes below, congruent triangles ABC and DEF are graphed.

Describe a sequence of rigid motions that maps triangle ABC onto triangle DEF.

Answer:


This is a "mean" question for two reasons. First, there is no reason to write a "sequence" because it can be done in one rotation. Second, it's set up in such a way to encourage a student to map ABC onto FED instead of DEF. Yes, that matters.

A rotation of 90 degrees counterclockwise around the origin would map ABC onto DEF.

A reflection over the y-axis and a translation of +1,+1 would map to the FED and receive half credit.

Under the new curriculum in NYS, you could also state, "Translate triangle ABC along ray AD until A comes to D. Then rotate ABC until point B coincides with point E."
(I'm not kidding.)




29. In triangle ADC below, EB is drawn such that AB = 4.1, AE = 5.6, BC = 8.22, and ED = 3.42.
Is triangle ABE similar to trianlge ADC? Explain why.

Answer:
There's a little bit of a hint when they didn't state "or why not".

Set up a proportion of corresponding sides. Remember to add the lines segments to get the lengths of AD and AC. We don't know anything about the lengths of BE or CD, but we do know that both triangles share angle A, which, of course, is congruent to itself because of the Reflexive Property.

Is 4.1 / (5.6 + 3.42) = 5.6 / (4.1 + 8.22) ?

Dividing, we get 0.4545... = 0.4545...

Or by cross-multiplying:

(5.6 + 3.42) (5.6) = 50.512; (4.1) (4.1 + 8.22) = 50.512

Since the corresponding sides are proportional, and because the two triangles share angle A, the triangles are similar by SAS.

If you didn't show the sides were propotional or you didn't mention angle A and SAS, you couldn't receive full credit.




30. Determine and state the coordinates of the center and the length of the radius of the circle represented by the equation x2 + 16x + y2 + 12y - 44 = 0.

Answer:
You have to complete the squares for both the x terms and the y terms to get the equation into the form (x - h)2 + (y - k)2 = r2, where (h,k) is the center of the circle and r is the radius of the circle. (Not r2 -- don't make that mistake!)

Has of 16 is 8, and 8 squared is 64. Half of 12 is 6, and 6 squared is 36. Add 64 and 36 to both sides of the equation. This means that (x + 8) and (y + 6) will both be in the final answer.

x2 + 16x + y2 + 12y - 44 = 0

x2 + 16x + 64 + y2 + 12y + 36 - 44 = 64 + 36

x2 + 16x + 64 + y2 + 12y + 36 = 100 + 44

(x + 8)2 + (y + 6)2 = 144

(x + 8)2 + (y + 6)2 = 122

The center of the circle is (-8,-6), and the radius is 12.

Both forget to "flip the signs" of the coordinates and to take the square root to find the radius.




31. In the diagram below, traingle SBC ~ triangle CMJ and cos J = 3/5.
Determine and state m∠S, to the nearest degree.

Answer:
This is another question that I do not like BECAUSE they hide important information. Since they stated that △SBC ~ △CMJ, then ∠S corresponds to ∠C, not ∠J.

This is necessary information. The problem is that in my years of teaching, many educators are a little lax in their naming conventions when stating the order of the vertices in the triangles. This could confuse many students. What's more since triangle SBC is the same as triangle BCS and triangle CBS, it isn't WRONG, per se, to write the letters in the incorrect order because it's still true.

So the entire question hinges on convention and whether or not the instructor followed it.

If you know enough about right triangles, you know enough to remember that if cos J = 3/5, then sin J = 4/5, and sin JCM = 3/5 and cos JCM = 4/5, because it is a multiple of a 3-4-5 triangle.

You can use trig ratios to find angles or Pythogean Theorem but that's a simple fact.

Since angle S correspond to angle C, then sin S = 3/5, and S = sin -1 (3/5) = 37 degrees.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



January 2024 Algebra 1 Regents Part II



This exam was adminstered in January 2024.

More Regents problems.

January 2024

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. Student scores on a recent test are shown in the table below.

85 96 92 82 90
90 88 95 85 88
90 87 96 82 85
92 96 85 92 87

On the number line below, create a dot plot to model the data

State the median test score for the data set.


Answer:


For each number in the table place a dot about that value on the number line. Make them all the same size so that, say, 3 dots above one number are the same height as 3 dots above another. Don't make any boxes or bars. You should have 20 dots when you are finished. If you have a different number of dots, you either left something out or repeated some piece of data.

Note that having exactly 20 dots is not a guarantee that you put them in the right place, but having 19 or 21 is definitely an error.

Your graph should look like this one:

There are 20 pieces of data, so the median test score will be the average of the 10th and the 11th. The 10th is 88 and the 11th is 90. The number in the middle of 88 and 90 is 89, which is the median.





26. State whether 2√(3) + 6 is rational or irrational. Explain your answer.

Answer:


2√(3) is an irrational number and the sum of a rational and an irrational number is always irrational.

If you mentioned that the decimal value goes on forever you must mention that it doesn't have a repeating pattern.





27. The table below shows data from a recent car trip for the Burke family.
Hours After Leaving (x) 1 2 3 4 5
Miles from Home (y) 45 112 178 238 305

State the average rate of change for the distance traveled between hours 2 and 4. Include appropriate units.


Answer:


Divide the difference of the miles at hour 4 and miles at hour 2 by the difference of 4 minus 2.

(238 - 112) / (4 - 2) = 63 mph.

It says to add appropriate units, so if you don't specify mph or miles per hour, you will lose a point.





28. On the set of axes below, graph the equation 3y + 2x = 15.

Explain why (-6,9) is a solution to the equation.

Answer:


Rewrite the equation into slope-intercept form to graph (or to put in your graphing calculator).

3y + 2x = 15
3y = -2x + 15
y = -2/3 x + 5

The slope is -2/3 and the y-intercept is 5. Start at (0,5) and use the rise (-2) and run (3) to find points on the line. Or use the table of values in the calclulator.

Your graph will look like this:

The point (-6,9) is a solution to the equation because it is a point on the line. All points on the line are solutions to the equation.





29.Using the quadratic formula, solve 3x2 - 2x - 6 = 0 for all values of x.
Round your answers to the nearest hundredth.

Answer:


Plug the values into the quadratic formula and evaluate. Don't forget to find two solutions.

Use a = 3, b = -2, and c = -6.

x = ( -b + √( (b)2 - 4(a)(c) ) / ( 2a )

x = ( -(-2) + √( (-2)2 - 4(3)(-6) ) / ( 2(3) )

x = ( 2 + √( 4 + 72) ) / (6)

x = ( 2 + √( 76) ) / (6) or x = ( 2 - √( 76) ) / (6)

x = 1.7862... or x = -1.1196...

x = 1.79 or x = -1.12





30. The piecewise function f(x) is given below.
f(x) = { 2x - 3, x > 3;
-x2 + 15, x < 3 }

State the value of f(3).
Justify your answer.

Answer:


Since 3 is not great than 3, but three is less than or equal to 3, use the second piece of the function.

f(3) = -(3)2 + 15 = -9 + 15 = 6

If you evaluate the top piece instead, you will get 1 credit.





31. Express the equation x2 - 8x = -41 in the form (x - p)2 = q.

Answer:


Complete the square by adding 16 to both sides. Half of -8 is -4 and (-4) square is 16.

x2 - 8x = -41
x2 - 8x + 16 = -41 + 16
(x - 4)2 = -25





32. Factor 36 - 4x2 completely.

Answer:


Remember that when it says "completely", there is usually more than one step. In this case there is a common factor of 4 in the two terms.

36 - 4x2
4(9 - x2)
4(3 - x)(3 + x)

The difference of two perfect squares always factors into two conjugates. (That is, two binomials with the same two terms, except that one has a + and the other has a -.)




End of Part II

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More to come. Comments and questions welcome.

More Regents problems.

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Order the softcover or ebook at Amazon.

Also, 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.

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Monday, February 05, 2024

August 2023 Algebra 2 Regents Part IV


This exam was adminstered in August 2023.

More Regents problems.

Algebra 2 August 2023

Part IV: A correct answer will receive 6 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


37.The Manford family started savings accounts for their twins, Abby and Brett, on the day they were born. They invested $8000 in an account for each child. Abby’s account pays 4.2% annual interest compounded quarterly. Brett’s account pays 3.9% annual interest compounded continuously. Write a function, A(t), for Abby’s account and a function, B(t), for Brett’s account that calculates the value of each account after t years.

Determine who will have more money in their account when the twins turn 18 years old, and find the difference in the amounts in the accounts to the nearest cent.

Algebraically determine, to the nearest tenth of a year, how long it takes for Brett’s account to triple in value.


Answer:


A is an exponential growth function. B is continuously compounded and uses e instead.

A(t) = 8000(1 + .042/4)4t
B(t) = 8000e.039t

For the second part, evaluate both functions for t = 18:

A(18) = 8000(1 + .042/4)4(18) = 16970.899 = $16,970.90

B(18) = 8000e.039(18) = 16142.273 = $16,142.27

The difference in the accounts will be 16970.90 - 16142.27 = $828.63.

To find when Brett's will triple in value, we need to find when B(t) = 24000.

8000e.039t = 24000
e.039t = 24000/8000
ln(e.039t) = ln(24000/8000)
.039t = 1.09861...
t = 28.1695...

In 28.2 years the value in Brett's account will triple.




End of Exam

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Available in softcover or ebook at Amazon.

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August 2023 Algebra 2 Regents, Part III


This exam was adminstered in August 2023.

More Regents problems.

Algebra 2 August 2023

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. 3 Sketch p(x) = -log2(x + 3) + 2 on the axes below.
Describe the end behavior of p(x) as x → -3.
Describe the end behavior of p(x) as x → ∞.

Answer:


Put the equation into your graphing calculator and look at the table of values. If you release that log2(1) is equal to 0, then you can deduce that p(-2) = 0 + 2, so (-2,2) is a point on the graph.

If you plot the points: (-2,2), (-1,1), (0,0.415), (1,0), (2,-0.322), and (3,-0.585), you'll have enough for a good sketch. There will be a vertical asymptote at x = -3.

Your graph should look like the following:

As x goes to -3, p(x) goes to infinity. As x goes to infinity, p(x) goes to negative infinity.





34. Solve for x algebraically: 1 / (x - 6) + x / (x - 2) = 4 / (x2 - 8x + 12)

Answer:


You need to combine the fractions on the left if you wish to cross-multiply. Or you can multiply both sides of the equation by (x2 - 8x + 12) if you realize that the polynomial factors into (x - 6)(x - 2).

1 / (x - 6) + x / (x - 2) = 4 / (x2 - 8x + 12)
(x - 6)(x - 2) ( 1 / (x - 6) + x / (x - 2) ) = (4 / (x2 - 8x + 12)) (x - 6)(x - 2)
x - 2 + x(x - 6) = 4
x - 2 + x2 - 6x = 4
x2 - 5x - 6 = 0
(x - 6)(x + 1) = 0
x - 6 = 0 or x + 1 = 0
x = 6 or x = -1

Throw out x = 6 as extraneous because 1 / (6 - 6) is undefined.

The only solution is x = -1.





35. Solve the following system of equations algebraically for x, y, and z.
2x + 4y - 3z = 12
3x - 2y + 2z = -9
-x + y - 3z = 0

Answer:


You can rewrite the third equation to solve for either x or y and then substitute the expression into the first two equations. Then you can solve that system.

-x + y - 3z = 0
y - 3z = x

2(y - 3z) + 4y - 3z = 12
3(y - 3z) - 2y + 2z = -9

2y - 6z + 4y - 3z = 12
3y - 9z - 2y + 2z = -9

6y - 9z = 12
y - 7z = -9

6y - 9z = 12
y = 7z - 9

6(7z - 9) - 9z = 12
42z - 54 - 9z = 12
33z = 66
z = 2

y = 7(2) - 9 = 5

x = y - 3z = (5) - 3(2) = -1





36. Two classes of students were entered into an experiment to see whether using an interactive whiteboard leads to better grades. It was observed that the mean grade of students in the class with the interactive whiteboard was 0.6 points higher than the class without it. To determine if the observed difference is statistically significant, the classes were rerandomized 5000 times to study these random differences in the mean grades. The output of the simulation is summarized in the histogram below.

Determine an interval containing the middle 95% of the simulation results. Round your answer to the nearest hundredth.
Does the interval indicate that the difference between the classes’ grades is significant? Explain.

Answer:


The interval is the mean plus or minus twice the standard deviation.

0.01 + 2(0.38) = 0.77
0.01 - 2(0.38) = -0.75

The interval is [-0.75, 0.77]

Since 0.6 is within the interval, so it's not significant.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.



Sunday, February 04, 2024

August 2023 Algebra 2, Part II


This exam was adminstered in August 2023.

More Regents problems.

Algebra 2 August 2023

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. Factor the expression 2x3 - 3x2 - 18x + 27 completely.

Answer:


Factor by grouping, and then factor the quadratic you get after the first step.

There are two ways to group, and either should work in any question of this kind.

2x3 - 3x2 - 18x + 27
(2x3 - 3x2) - (18x - 27)
x2(2x - 3) - 9(2x - 3)
(x2 - 9)(2x - 3)
(x + 3)(x - 3)(2x - 3)

You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.

2x3 - 18x - 3x2 + 27
(2x3 - 18x)(3x2 - 27)
2x(x2 - 9) - 3(x2 - 9)
(2x - 3)(x2 - 9)
(2x - 3)(x + 3)(x - 3)





26. Algebraically determine the values of x that satisfy the system of equations shown below:

y = x2 + 8x - 5
y = 8x - 4

Answer:


Set the two statements equal to each other. Solve the quadratic equation for x. You don't need to find y.

y = x2 + 8x - 5
y = 8x - 4

x2 + 8x - 5 = 8x - 4
x2 - 1 = 0
(x - 1)(x + 1) = 0
x - 1 = 0 or x + 1 = 0
x = 1 or x = -1





27. Solve the equation 3x2 + 5x + 8 = 0. Write your solution in a + bi form.

Answer:


When they tell you "a + bi" form, you know that it won't be something that you can factor easily and that you need to use the quadratic formula.

3x2 + 5x + 8 = 0

x = (-b + √(b2 - 4ac) ) / (2a)

x = (-5 + √(52 - 4(3)(8)) ) / (2(3))

x = (-5 + √(25 - 96) ) / (6))

x = (-5 + √(-71) ) / (6))

x = -5/6 + i √(71)/6





28. On the coordinate plane below, sketch at least one cycle of a cosine function with a midline at y = -2, an amplitude of 3, and a period of π/2.

Answer:


Cosine doesn't start at the midline. Cosine starts at the midline plus the amplitude, which in this case is -2 + 3 = 1. The minumim will occur at y = -2 - 3 = -5 when x = (1/2)(π/2) = π/4.

Your sketch should look like this:

Remember that a sketch doesn't have to be perfect, but it shouldn't have any obvious errors, such as crossing the axis at the wrong place.





29. Given i is the imaginary unit, simplify (5xi3 - 4i)2 as a polynomial in standard form.

Answer:


Use FOIL, the Distributive Property or the Box Method/Area Model. Remember that i2 = -1, i3 = -i, i4 = 1, i5 = i, and so forth.

(5xi3 - 4i)2
25x2i6 - 40xi4 + 16i2
-25x2 - 40x - 16





30. Consider the parabola given by y = 1/4 x2 + x + 8 vertex (-2,7) and focus (-2,8). Use this information to explain how to determine the equation of the directrix.

Answer:


The focus is 1 unit above the vertex, so the directrix is the horizontal line that contains the point 1 unit below the vertex, (-2,6).

The equation would be y = 6.

Remember that you have to give an explanation, not just the equation. The equation alone is half-credit.





31.Write (x √(x3) ) / (∛(x5)

Answer:


Remove the radicals by rewriting the expressions with fractional exponents. Then using the laws of exponents subtract the exponent of the numerator from the exponent of the denominator.

(x √(x3) ) / (∛(x5)

x (x (3/2)) / x (5/3)

x (5/2) / x (5/3)

x (5/2 - 5/3)

x (15/6 - 10/6)

x (5/6)





32. A fruit fly population can be modeled by the equation P = 10(1.27)t, where P represents the number of fruit flies after t days. What is the average rate of change of the population, rounded to the nearest hundredth, over the interval [0,10.5]? Include appropriate units in your answer.

Answer:


Find P(0) and P(10.5), subtract the latter from the initial value and then divide by 10.5.

( 10(1.27)10.5 - 10(1.27)0) / (10.5 - 0) = 10.7628...

10.76 fruit flies per day.




End of Part II

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, 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.