Tuesday, September 10, 2019

Palindrome Dates

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

I shy away from a lot of obvious date jokes, but it's the last palindrome week of the century, so I don't think I'll be doing it again!




Come back often for more funny math and geeky comics.




Friday, September 06, 2019

Summer Expectations

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

Jump to aqua-space on my mark.

I wanted a beat panel, followed by uncontrollable laughter, and then sliding under, but it was too space. So I went with Ackbar.

This image was created on Saturday 9/7 for Friday's strip. It was late, and I've fallen behind a bit.
Hopefully, Monday's strip will appear on Monday.

Actually, this would have been a good, ahem, Labor Day strip, if only because three of them went back to work right afterward.




Come back often for more funny math and geeky comics.




Wednesday, September 04, 2019

Summer Travel

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

I wanted to say it was a "Thai" for the best summer story, but I didn't have any others to run with.

This image was created on Friday 9/6 for Wednesday's strip. It was late, and I've fallen behind a bit.
Hopefully, Friday's strip will be created on Friday as well.




Come back often for more funny math and geeky comics.




Tuesday, September 03, 2019

August 2019 Geometry Regents Part 1 (multiple choice)

The following are some of the multiple questions from the recent June 2019 New York State Common Core Geometry Regents exam.

June 2019 Geometry, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.


1. On the set of axes below, AB is dilated by a scale factor of 5/2 centered at point P.


Which statement is always true?

Answer: (2) AB | A'B'
Shape (and therefore slope) is preserved in a dilation, so the line segments are parallel.
Choice (4) is backward. If it had been 5/2(AB) = A'B', that would have been correct.


2. The coordinates of the vertices of parallelogram CDEH are C(-5,5), D(2,5), E(-1,-1), and H(-8,-1). What are the coordinates of P, the point of intersection of diagonals CE and DH?

Answer: (3) ( 3,2)
In a parallelogram, the diagonals bisect each other, so the point of intersection must be the midpoint of both lines.
The midpoint of CE is at ( (-5+-1)/2, (5+-1)/2 ), which is (-3, 2).
Checking the other line, just to be sure:
The midpoint of DH is at ( (2+-8)/2, (5+-1)/2 ), which is (-3, 2).


3. The coordinates of the endpoints of QS are Q(-9,8) and S(9,-4). Point R is on QS such that QR:RS is in the ratio of 1:2. What are the coordinates of point R?

Answer: (3) (-3,4)
A ratio of 1:2 means one-third of the way along he line.
The difference in the x-coordinates is 9 - (-9) = 18. One-third of 18 is 6, and -9 + 6 = -3.
The eliminates all but one choice.
Check the y-coordinates to be sure:
The difference in the y-coordinates is -4 - 8 = -12. One-third of -12 is -4, and 8 + -4 = 4.


4. If the altitudes of a triangle meet at one of the triangle’s vertices, then the triangle is

Answer: (1) a right triangle
There's no way for this to happen unless one of the angles is a right angle, which makes it a right triangle.


5. In the diagram below of triangle ACD, DB is a median to AC, and AB = DB
If m∠DAB = 32°, what is m∠BDC?

Answer: (3) 58°
If m∠DAB = 32°, then m∠ADB = 32° because it is an isosceles triangle.
This makes m∠DBC = 64° by the Remote Angle Theorem.
Since B is a median, AB = BC, but since AB = DB, then BC = DB, making DBC an isosceles triangle.
This makes BDC = BCD, and the pair of base angles have a sum equal to 180 - 64 = 116.
Half of 116 is 58, so m∠BDC = 58°


6. What are the coordinates of the center and the length of the radius of the circle whose equation is x2 + y2 = 8x - 6y + 39?

Answer: (2) 128
To put is into standard form (x - h)2 + (y - k)2 = r2, you need to move the x and y terms to the left side and then Complete the Square, twice.
Remember, you need to take half of the coefficient of x (or y) and then square it to find out what to add.

x2 + y2 = 8x - 6y + 39
x2 - 8x + y2 + 6y = 39



7. The line 3x 4y 8 is transformed by a dilation centered at the origin. Which linear equation could represent its image?

Answer: (2) y = 3/4 x + 8
A line that has been dilated would keep its slope and be parallel to the original (unless it was the same line).
The slope of the original line can be found by isolating y.
-3x + 4y = 8
4y = 3x + 8
y = 3/4 x + 8

The slope must be 3/4, so the choice is (2).

Also, when written in Standard Form Ax + By = C, the slope of the line is -A/B, which in this case is -3/-4 = 3/4.


8. In the diagram below, AC and BD intersect at E.
Which information is always sufficient to prove triangle ABE = triangle CDE?

Answer: (4) BD and AC bisect each other.
Bisecting each other means that the two triangles will be congruent because of SAS, usually the vertical angles between the sides.
Choice (1) is enough to prove that they are similar by AAA because of Alternate Interior Angles, but not that they are congruent.
Choice (2) gives you SSA, which isn't a postulate or theorem. (And don't read it backward!)
Choice (3) isn't enough information. You'll have one side and one angle.


9. The expression sin 57° is equal to

Answer: (2) cos 33°
The sine of an angle (n) is equal to the cosine of the complementary angle (90 - n).


10. What is the volume of a hemisphere that has a diameter of 12.6 cm, to the nearest tenth of a cubic centimeter?

Answer: (1) 523.7
Volume of a sphere is V = (4/3) pi * r3
A hemisphere is half of that. The radius is half of the 12.6 diameter, or 6.3
So V = (1/2)(4/3)(3.141592)(6.3)3 = 523.6971...

A quick guess of the wrong answers would be (2) forget the 1/2, (3) took 1/2 the volume but used the diameter, (4) made both mistakes.


11. In the diagram below of triangle ABC, D is a point on BA, E is a point on BC, and DE is drawn.
If BD = 5, DA = 12, and BE = 7, what is the length of BC so that AC || DE?

Answer: (1) 23.8
For the lines to be parallel, the following proportion needs to be true. Notice that it says EC and NOT BC.
BD / DA = BE / EC
5 / 12 = 7 / EC
5 EC = 84
EC = 16.8
BC = BE + EC = 7 + 16.8 = 23.8

You could have done BD/BA = BE/BC if you preferred.


12. A quadrilateral must be a parallelogram if

Answer: (3) one pair of sides is both parallel and congruent
If the lines are parallel and congruent, then the quadrilateral must be a parallelogram.
Choices (1), (2) and (4) could be an isosceles trapezoid.


13. In the diagram below of circle O, chords JT and ER intersect at M.


If EM = 8 and RM = 15, the lengths of JM and TM could be

Answer: (3) 16 and 7.5
If two chords of a circle intersect, then the products of their segments will be equal.
So (JM)(MT) = (RM)(ME) = (15)(8) = 120
Only Choice (3) has two factors with a product of 120.


14. Triangles JOE and SAM are drawn such that ∠E = ∠M and EJ = MS. Which mapping would not always lead to triangle JOE = triangle SAM?

Answer: (4) JO maps onto SA

Choice (1) gives you ASA. Choice (2) gives you AAS. Choice (3) gives you SAS. Choice (4) gives you SSA, which is not a theorem nor postulate (as stated in an earlier question).


15. 5 In triangle ABC shown below, ∠ACB is a right angle, E is a point on AC, and ED is drawn perpendicular to hypotenuse AB.
If AB = 9, BC = 6, and DE = 4, what is the length of AE?

Answer: (2) 6
The triangles are similar because they each have a right angle and they share angle A. That means that the corresponding sides are proportional. However, be careful not to mix up the sides because of the way it is drawn. AE is the hypotenuse of the smaller triangle.

AB / BC = AE / DE
9 / 6 = AE / 4
6 AE = (9)(4) = 36
AE = 6


16. Which equation represents a line parallel to the line whose equation is -2x + 3y = -4 and passes through the point (1,3)?

Answer: y - 3 = (2/3)(x - 1)

The choices are all in Point-slope form, which means y - 3 = m(x - 1), where m is the same slope as the original equation.
Immediately, cross off choices (3) and (4).
Spoiler alert: the slope is going to be positive, so (1) is wrong, too, but let's continue.

-2x + 3y = -4
3y = 2x - 4
y = 2/3x - 4, slope = 2/3.

As stated in an earlier question, when the equation is in Standard Form Ax + By = C, the slope is -A/B.
So the slope was -(-2)/3 = 2/3.


17. In rhombus TIGE, diagonals TG and IE intersect at R. The perimeter of TIGE is 68, and TG = 16.


What is the length of diagonal IE?

Answer: (2) 30
They chose the numbers carefully. There are four congruent right triangles making up that rhombus.
The perimeter is 68, meaning that each side is 17, which is the hypotenuse of the triangles. TG = 16, but since TG and IE bisect each other, TR and RG are each 8. This is one leg of the right triangles.
So 82 + ER2 = 172
If you've followed my advice before this, you've learned from Pythagorean Triples and know that these are 8-15-17 triangles.
If you didn't ... sigh, okay, let's do it:

82 + ER2 = 172
64 + ER2 = 289
ER2 = 225
ER = 15

ER = 15 = IR, so IE = 30.


18. In circle O two secants, ABP and CDP, are drawn to external point P. If mAC = 72°, and mBD = 34°, what is the measure of ∠P?

Answer: (1) 19°
Angle P will be one-half of the difference between the two arcs.
So (72° - 34°) / 2 = 19°.


19. What are the coordinates of point C on the directed segment from A(-8,4) to B(10,-2) that partitions the segment such that AC:CB is 2:1?

Answer: (4) (4,0)
To get a 2:1 ratio, C has to be 2/3 of the way along the line from A to B.
To get from -8 to 10, you have to move 18 units. Multiply 2/3 * 18 = 12. Add 12 to -8 to get 4.
To get from 4 to -2, you have to move 6 units. Multiply 2/3 * 6 = 4. Subtract 4 from 4 to get 0.
Subtract in the second case because you are moving down.


20. The equation of a circle is x2 + 8x + y2 - 12y = 144. What are the coordinates of the center and the length of the radius of the circle?

Answer: (4) center (-4,6) and radius 14
You need to Complete the Square. (Yes, that's back from Algebra).
Half of 8 is 4, and 42 is 16
Half of -12 is -6, and (-6)2 is 36.
x2 + 8x + y2 - 12y = 144
x2 + 8x + 16 + y2 - 12y + 36 = 144 + 16 + 36
(x + 4)2 + (y - 6)2 = 196
The center is (-4, 6) and the radius is SQRT(196) = 14.


21. In parallelogram PQRS, QP is extended to point T and ST is drawn.


If ST = SP and m∠R = 130°, what is m∠PST?

Answer: (2) 80°
There is a parallelogram and an isosceles triangle.
Angle QPS = angle R = 130 degrees. Angle SPT is supplementary to QPS, so it is 50 degrees. Since PST is an isosceles triangle, the base angles are equal, so angle T is also 50 degrees. The vertex angle PST is 180 - 50 - 50 = 80 degrees.


22. A 12-foot ladder leans against a building and reaches a window 10 feet above ground. What is the measure of the angle, to the nearest degree, that the ladder forms with the ground?

Answer: (4) 56
Before starting, realize that the angle with the ground must be bigger than the angle with the building, which means it has to be over 45 degrees. So you can eliminate choices 1 and 2.
The wall is opposite the angle, and the ladder is the hypotenuse, so you need to use Sine to find the angle.
Sin x = opp/hyp
Sin x = (10/12)
x = sin-1(10/12) = 56.44...

If you got a ridiculously low decimal, then your calculator is in Radians mode.


23. In the diagram of equilateral triangle ABC shown below, E and F are the midpoints of AC and BC, respectively.


If EF = 2x + 8 and AB = 7x - 2, what is the perimeter of trapezoid ABFE?

Answer: (3) 100
Since EF is a midsegment, then it is half the length of AB.
Since ABC is an equilateral triangle, EA and FB are half the length of AB.

So first we need to find x and then EF.

2(2x + 8) = 7x - 2
4x + 16 = 7x - 2
16 = 3x - 2
18 = 3x
x = 6

Since x = 6, then EF = 2(6) + 8 = 12 + 8 = 20.
The perimeter is equal to FIVE TIMES the length of EF, so 20 * 5 = 100.

Where did I get FIVE TIMES from? EF = EA = FB = 1/2(AB), so AB = 2 EF.


24. Which information is not sufficient to prove that a parallelogram is a square?

Answer: (3) The diagonals are perpendicular and one pair of adjacent sides are congruent.
Choice (3) describes a rhombus which does not have to be a square. Each of the other choices have one condition that makes it a rhombus and another that makes it a rectangle. If it is both a rhombus and a rectangle, it must be a square.

End of Part I

How did you do?

Questions, comments and corrections welcome.

Friday, August 30, 2019

August 2019 Geometry Regents Parts 3 & 4

The following are some of the multiple questions from the August 2019 New York State Geometry Regents exam.

August 2019 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.


32. Triangle ABC is shown below. Using a compass and straightedge, construct the dilation of Triangle ABC centered at B with a scale factor of 2. [Leave all construction marks.]

Is the image of triangle ABC similar to the original triangle? Explain why.

Answer:
Starting at point B, open the compass to Point A. Then from point A, make an arc. Use the straightedge to extend BA to the new arc. This line will have twice the length of BA.
Repeat the process for BC.
Finally, using the straightedge, complete the triangle from the endpoints of the two new line segments you constructed.

The image is similar to the original triangle because a dilation preserves the shape of the object. Therefore, the angles are the same size and by the AA Theorem, the two triangles are similar.


33. In the diagram below, triangle ABE = triangle CBD.

Prove: triangle AFD = triangle CFE

Answer:
You need to prove that those two tiny triangles are similar.
You can see that Angles A and C are equal from the original triangles. And angles AFD and CFE are vertical angles. You need to have one pair of congruent sides.

You don't know that AD = CE directly, but you do know that AB = CB and DB = EB because of the congruency of the larger triangles. Therefore if you subtract the smaller segment from the larger one, you get the sides that we need to be congruent.

Your proof would go something like this:

StatementReason
1. Triangle ABE = Triangle CBD1. Given
2. Angle A = Angle C2. CPCTC
3. Angle AFD = Angle CFE3. Vertical angles are congruent.
4. AB = CB; DB = EB4. CPCTC
5. AD = CE5. Subtraction Postulate
6. triangle AFD = triangle CFE/td>6. AAS



34. A cargo trailer, pictured below, can be modeled by a rectangular prism and a triangular prism. Inside the trailer, the rectangular prism measures 6 feet wide and 10 feet long. The walls that form the triangular prism each measure 4 feet wide inside the trailer. The diagram below is of the floor, showing the inside measurements of the trailer.


If the inside height of the trailer is 6.5 feet, what is the total volume of the inside of the trailer, to the nearest cubic foot?

Answer:
Volume = Area of the Base times the Height.
The base is shaped like a pentagon, which, in this case, can be broken into a rectangle and a triangle. We have the base of the triangle, 6 feet; however, we need to solve for its height (lower case h). We can use the Pythagorean Theorem for this.

32 + x2 = 42
9 + x2 = 16
x2 = 7
x = SQRT(7) = 2.65 (approximately)
Area of the base = (6)(10) + (1/2)(6)(2.65) = 67.95
Volume = Area * height = 67.95 * 6.5 = 441.675 = 442 cubic feet


Part IV

A correct answer is worth up to 6 credits. Partial credit is available.

35. The coordinates of the vertices of Triangle ABC are A(1,2), B(-5,3), and C(-6,-3).

Prove that Triangle ABC is isosceles
[The use of the set of axes on the next page is optional.]

State the coordinates of point D such that quadrilateral ABCD is a square.
Prove that your quadrilateral ABCD is a square.
[The use of the set of axes below is optional.]

Answer:
Two separate questions.
First, to prove that ABC is isosceles, you will have to show that two of the sides have the same length.
It can be assumed from the second part of the question even before we start that it will be a right triangle, because that is the only way that you will be able to add one more point and have a square. Furthermore, AB and BC would have to be the congruent sides.

The second and third parts are really the same question. Once you have the square, you have to prove that it is a square. This is more rigorous than just an "explanation" or "justification". However, by this point you have already shown that two consecutive sides are congruent, so all you have to do is show that the sides are parallel, making it both a parallelogram and a rhombus, and that there is at least one right angle, which makes it a square.

First Part:
Length of AB = SQRT( (-5 - 1)2 + (3 - 2)2 ) = SQRT(37)
Length of BC = SQRT( (-6 - -5)2 + (-3 - 3)2 ) = SQRT(37)
AB = BC, so triangle ABC is isosceles.
(No reason to check the length of AC.

Second Part:
To get from point A to B, you have to go down 6 left and up 1. (This can be seen if you graph it, or if you worked out the math above.) CD would have to have the same slope, but to get from C to D, you would have to go 6 right and down 1. So D would have to be (-6 + 6, -3 - 1) or D(0, -4).

Third part:
The slope of AB = -1/6. The slope of CD = -1/6
The slope of BC = (-3 -3)/(-6 - -5) = -6/-1 = 6
The slope of DA = (-4 - 2)/(0 - 1) = -6 / -1 = 6

AB || CD, AD || BC because they have the same slopes.
AB is perpendicular to BC because their slopes are inverse reciprocals (have a product of -1).
So Angle B is a right angle.

A quadrilateral with opposite sides parallel, with consecutive sides equal and a right angle must be a square.

End of Exam

How did you do?
Comments and corrections welcome. (I get many of the latter!)

Thursday, August 29, 2019

Numberpalooza

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

Apparently, many (x, why?) songs are sung by shapes or symbols. And most of my Zero characters were *not* singers.

I thought it was going to be a few zeroes and tens when I started going through the old comics. And I never did, say, Boy Four-ge, even though a song was used for a truth table.




Come back often for more funny math and geeky comics.




Wednesday, August 28, 2019

August 2019 Geometry Regents, Part 2

The following are some of the multiple questions from the recent August 2019 New York State Common Core Geometry Regents exam.

August 2019 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 parallelogram ABCD shown below, m∠DAC = 98° and m∠ACD = 36°.


What is the measure of angle B? Explain why.

Answer:
The opposite angles in a parallelogram are congruent, so m∠B = m∠D. Also, ACD is a triangle, so the sum of its angles is 180 degrees.
180 - 98 - 36 = 46 degrees.
Angle D is 46 degrees, so angle B is 46 degrees.


26. An airplane took off at a constant angle of elevation. After the plane traveled for 25 miles, it reached an altitude of 5 miles, as modeled below.


To the nearest tenth of a degree, what was the angle of elevation?

Answer:
The angle of elevation is measured from the ground. That makes the altitude of 5 miles the side opposite of the angle, and the 25 miles is the hypotenuse. This means you need to use sine.
Sin x = 5/25
x = sin-1 (1/5) = 11.53...
The angle of elevation is 11.5 degrees, to the nearest tenth.


27. On the set of axes below, Triangle ABC = Triangle DEF.


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

Answer:
Some possibilities

A 180 degree rotation of ABC about the origin, followed by a translation of 4 spaces up.

A reflection of ABC across the y-axis, followed by a reflection across the line y = 2.

A reflection of ABC across the y-axis, followed by a reflection across the x-axis, followed by a translation 4 spaces up.

A 180 degree rotation of ABC about the point (0, 2).

You could also pick any point A, B, or C, and translate ABC so that it moves the point onto the corresponding point of DEF, and then rotate 180 degrees about that corresponding point. (You need to fill in the specifics -- the previous sentence by itself will not give you credit.)


28. The vertices of Triangle ABC have coordinates A(-2,-1), B(10,-1), and C(4,4). Determine and state the area of Triangle ABC. [The use of the set of axes below is optional.]

Answer:
Since A and B both have a y-coordinate of -1, you know that AB is a horizontal line and you can use that as the base of your triangle. The height of the triangle is the vertical distance from -1 to 4. And the area of the triangle is 1/2 (base) (height).

b = 10 - (-2) = 12
h = 4 - (-1) = 5
A = 1/2(12)(5) = 30

Note that 72/360 is 1/5, which after a year of Geometry, you should probably recognize. One-fifth of ten squared is 1/5 of 100, which is 20.


29. Using the construction below, state the degree measure of ∠CAD. Explain why.

Answer:
A reverse construction question! I like it!

Triangle ABC is an equilateral triangle because sides AC and BC are constructed to be the same length of AB.
Ray AD is constructed to be a bisector of angle CAB.
Since ABC is equilateral, m∠CAB = 60 degrees. Therefore, m∠CAD = 30 degrees, which is half of ∠CAB.

Note: You could have used a less wordy explanation than that, but I like to explain.


30. In the diagram below of circle K, secant PLKE and tangent PZ are drawn from external point P.


If m LZ = 56°, determine and state the degree measure of angle P.

Answer:
K is the center of the circle, so LE is a diameter splitting the circle into two semicircles.
This means that arc EZL has a measure of 180 degrees, of which arc LZ is 56 degrees.
Therefore, arc EZ is 180 - 56 = 124 degrees.

The measure of angle P can be found by taking half of the difference between arc EZ and LZ:
1/2 (124 - 56) = 1/2 (68) = 34 degrees.

Alternatively, because arc LZ measures 56 degrees, then central angle LKZ has a measure of 56 degrees. If you draw radius KZ, it will be perpendicular to tangent PZ. That means that PKZ is a right triangle and you already know the size of two of its angles.
Angle P + 56 + 90 = 180, so Angle P = 34 degrees.


31. A large water basin is in the shape of a right cylinder. The inside of the basin has a diameter of 8 1/4 feet and a height of 3 feet. Determine and state, to the nearest cubic foot, the number of cubic feet of water that it will take to fill the basin to a level of 1/2 foot from the top.

Answer:
You want to find the volume of the cylinder, but to a height of 2 1/2 feet instead of 3 feet. The radius is half of the diameter, which is (1/2)(8 1/4) = 4 1/8 Use the pi key on your calculator to get extra decimals for pi. Do NOT use 3.14 because it will not be accurate enough.
V = pi * r2 * h
V = 3.141592... * (4.125)2 * 2.5 = 133.64 = 134 cubic feet.

End of Part II

How did you do?

Questions, comments and corrections welcome.

Monday, August 26, 2019

August 2019 Algebra 1 Regents Parts 3 & 4

The following are some of the multiple questions from the August 2019 New York State Common Core Algebra I Regents exam.

August 2019 Algebra I, Part III

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


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

2x + y > 8
y - 5 < 3x

Determine if the point (1,8) is in the solution set. Explain your answer.

Answer:
Rewrite the equations in slope-intercept form

y > -2x + 8
y - 5 < 3x + 5

Note that the first is greater than or equal to, so it will be a solid line that is shaded above.
The second is less than, so it will be a broken or dashed line that is shaded below.
If you are not sure which side to shade, substitute the point (0, 0) in the inequality. If you get a true statement as a result, then shade that side. If the statement is false, shade the other side.
IMPORTANT: this test doesn't work if the line goes through (0, 0). In that case, pick any other point not on the line. It will work for any point, but (0, 0) is the easiest one to use!

The first line (solid) has a slope of -2 and a y-intercept of 8.
The second line (broken) has a slope of 3 and a y-intercept of 5.


The point (1, 8) is NOT part of the Solution Set because it is on the broken line, which is a boundary, but not a part of the solution.
Important: if you have a graphing error, then your response to this part of the question should match your graph to get the point.


34. On the day Alexander was born, his father invested $5000 in an account with a 1.2% annual growth rate. Write a function, A(t), that represents the value of this investment t years after Alexander’s birth.

Determine, to the nearest dollar, how much more the investment will be worth when Alexander turns 32 than when he turns 17.

Answer:
For the first part, you have to write a formula.
For the second part, you have to use the formula twice for when he is 32 and 17, and subtract the difference.

A(t) = 5000 (1.012)t

A(32) - A(17) = 5000 (1.012)32 - 5000 (1.012)17 = 1199.91758695
$1200.

You can put that entire equation in your calculator. You do not need to evaluate the parts separately.


35. Stephen collected data from a travel website. The data included a hotel’s distance from Times Square in Manhattan and the cost of a room for one weekend night in August. A table containing these data appears below.

Distance From
Times Square
(city blocks) (x)
0 0 1 1 3 4 7 11 14 19
Cost of
a Room
(dollars) (y)
293 263 244 224 185 170 219 153 136 111

Write the linear regression equation for this data set. Round all values to the nearest hundredth.

State the correlation coefficient for this data set, to the nearest hundredth.

Answer:
Enter the data into two lists, L1 and L2.
Make sure DiagnosticOn has been set. (If you don't know, just do select it from the Menu.)
Press STAT to get to the statistics menu and arrow over to CALC. Choose option 4, LinReg(ax+b).
You will get a = -7.76, b = 246.34 and r = -.88, rounded to the nearest hundredth.

The linear regression is y = -7.76x + 246.34

The correlation coefficient is r = -0.88

In the context of this problem, the correlation coefficient suggest a strong negative correlation between the number of blocks from Times Square and the cost of a room. As you get farther away from Times Square, the cost of the room decreases.
(It's not enough to just say that it's a strong negative correlation.)

Coincidentally, this question is very similar to Question 35 of the June 2019 Regents. I barely had to do any editing to my solution (which, yes, I cut and pasted from the old post because all the formatting is in place).


36. A snowstorm started at midnight. For the first 4 hours, it snowed at an average rate of one-half inch per hour.
The snow then started to fall at an average rate of one inch per hour for the next 6 hours.
Then it stopped snowing for 3 hours.
Then it started snowing again at an average rate of one-half inch per hour for the next 4 hours until the storm was over.
On the set of axes below, graph the amount of snow accumulated over the time interval of the storm.

Determine the average rate of snowfall over the length of the storm. State the rate, to the nearest hundredth of an inch per hour.

Answer:
Notice that the scale for the x-axis is 1 box = 1 hour, but the scale for the y-axis is 1 box = 1/2 inch of snow. Don't let that confuse you.
You can plot points and draw line segments just by counting the boxes. The thing to remember is if no snow falls, the accumulation doesn't change. Slope is 0 (flat line) for that time period.
To find the average, what would the slope be between the origin (0,0) and the highest point on the graph at the end of the storm, which should be (17, 10)?
(10 - 0) / (17 - 0) = 10 / 17 = 0.5882... = 0.59 inches per hour (to the nearest hundredth).

Again, if you made a graphing error, use whatever point you finished at.
For example, if you missed a point when it stopped for 3 hours (suppose you graphed 2), and you ended up at (16, 10), you would have done 10/16 = 0.625, which is 0.63. You get credit if you are consistent.

August 2018 Algebra I, Part IV

A correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.


37. Allysa spent $35 to purchase 12 chickens. She bought two different types of chickens. Americana chickens cost $3.75 each and Delaware chickens cost $2.50 each.

Write a system of equations that can be used to determine the number of Americana chickens, A, and the number of Delaware chickens, D, she purchased.

Determine algebraically how many of each type of chicken Allysa purchased.

Each Americana chicken lays 2 eggs per day and each Delaware chicken lays 1 egg per day. Allysa only sells eggs by the full dozen for $2.50. Determine how much money she expects to take in at the end of the first week with her 12 chickens.

Answer:
A + D = 12
3.75A + 2.50D = 35

You can solve by Sustitution (Substitute 12 - A for D) or Elimination (multiply the first equation by -2.50).


A + D = 12
3.75A + 2.50D = 35
3.75A + 2.50(12 - A) = 35
3.75A + 30 - 2.50 = 35
1.25A + 30 = 35
1.25A = 5
A = 4
4 + D = 12
D = 8
4 American and 8 Delaware

Or

A + D = 12
3.75A + 2.50D = 35
-2.50A - 2.50D = 30
1.25A + 30 = 35
1.25A = 5
A = 4
4 + D = 12
D = 8
4 American and 8 Delaware

The number of eggs laid per day are 4(2) + 8(1) = 16.
After one week, she will have 16 * 7 = 112 eggs
This makes 112 / 12 = 9.333.... dozen.
She can sell 9 dozen for $2.50 each, which is 9 * 2.50 = $22.50.


I would hate to have to follow through the work, checking for consistency, if the number of chickens was incorrect!




End of Exam

How did you do?

Questions, comments and corrections welcome.

Sunday, August 25, 2019

August 2019 Algebra 1 Regents, Part 2

The following are some of the multiple questions from the August 2019 New York State Common Core Algebra I Regents exam.

August 2019 Algebra I, Part II

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


25. If g(x) = -4x2 - 3x + 2, determine g(2).

Answer:
Substituion:
g(2) = -4(-2)2 - 3(-2) + 2
= -4(4) - (-6) + 2
= -16 + 6 + 2 = -8

One point off for one computational error, such as (-2)2 = -4, or similar.


26. A student is in the process of solving an equation. The original equation and the first step are shown below.


Original: 3a + 6 = 2 - 5a + 7
Step one: 3a + 6 = 2 + 7 - 5a
Which property did the student use for the first step? Explain why this property is correct.

Answer:
The student used the Commutative Property to switch the order of the last to terms, which allows the student to Combine Like Terms in the next step.


27. On the set of axes below, graph the line whose equation is 2y = -3x - 2.
This linear equation contains the point (2,k). State the value of k

Answer:
Divide the equation by 2 to get it in slope-intercept form. You will see that the y-intercept is -1 and the slope is -3/2. You could use the graphing calculator at this point, or you can plot the point (0, 11) and then make you next point down 3 spaces and 2 to the right, and then repeat. When you get to the edge of the graph, go back the other way: 3 spaces up and 2 to the left, and repeat. And the arrows.

You can plainly see that the point (2, k) is (2, -4), so k = -4.

If you graphed this incorrectly, but your answer for k matches your graph, you will get credit for that portion.
If you used the equation 2y = -3(2) - 2 to find the y-coordinate, you will probably get the credit.




28. The formula a = (vf - vi) / t is used to calculate acceleration as the change in velocity over the period of time.
Solve the formula for the final velocity, vf, in terms of initial velocity, vi, acceleration, a, and time, t.

Answer:
This is a real-world problem: calculating a final velocity when given an initial velocity, acceleration and a period of time.
Isolate the variable by moving everything else to the other side of the equation.

a = (vf - vi) / t
at =vf - vi
at + vi = vf


29. Solve 3/5 x + 1/3 < 4/5 x - 1/3 for x.

Answer:
You can eliminate the fractions, if you want, by multiplying the entire equation by 15. Or you can simplify it a little first.

3/5 x + 1/3 < 4/5 x - 1/3
2/3 < 1/5 x
(15)(2/3) < (1/5 x)(15)
10 < 3x
10/3 < x


30. Is the product of two irrational numbers always irrational? Justify your answer.

Answer:
No, sometimes the answer is rational. SQRT(2) * SQRT(2) = 2, which is rational.


31. Solve 6x2 - 42 = 0 for the exact values of x.

Answer:
Exact values of x means that you do NOT estimate irrational numbers. Do not round anything.

6x2 - 42 = 0
6x2 = 42
x2 = 7
x = +SQRT(7)

If you go any further than that, you will lose a point, so STOP.

If you used the Quadratic Formula, you would have gotten something like + SQRT(1008) / 12. This is OKAY because that is an Exact value, which can be simplified into 12 * SQRT(7) / 12, but you wouldn't be required to do this. However, if you did attempt to simplify it, you would have to get it right.


32. Graph the function:

h(x) = { 2x - 3, x < 0
x2 - 4x - 5, 0 < x < 5

Answer:
ONLY Graph up to x = 5 and no farther. No arrow. The left side, however, does get an arrow. The left portion ends with an OPEN circle. The parabola starts with a CLOSED circle. You could end it with a closed circle as well, if you want, or just a point.

End of Part II

How did you do?

Questions, comments and corrections welcome.

Friday, August 23, 2019

August 2019 Algebra 1 Regents Part 1

The following are some of the multiple questions from the recent August 2019 New York State Common Core Algebra I Regents exam.

Omitted images will be added soon.

August 2019 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.


1. Bryan’s hockey team is purchasing jerseys. The company charges $250 for a onetime set-up fee and $23 for each printed jersey. Which expression represents the total cost of x number of jerseys for the team?

Answer: (3) 23x + 250
The start-up fee is paid once, so it doesn't get a variable. It's the cost even if 0 jerseys are made, so it's the y-intercept of a graph. The $23 is a repeated cost per jersey, so it gets a variable.


2. Which table represents a function?

Answer: (4)


Choices (1), (2) and (3) each repeat an x value with a different y value. That is not allowed in a function.


3. Which expression is equivalent to 2(x2 - 1) + 3x(x - 4)?

Answer: (4) 5x2 - 12x - 2


2(x2 - 1) + 3x(x - 4)
2x2 - 2 + 3x2 - 12x
5x2 - 12x - 2
You might've used process of elimination part-way through this problem.


4. The value of x that satisfies the equation 4/3 = (x + 10) / 15 is

Answer: (3) 10
The quick way: 3 goes into 15 five times, and 4 * 5 = 20 = x + 10. Then x = 10.
or


4/3 = (x + 10) / 15
(3)(x + 10) = (4)(15)
3x + 30 = 60
3x = 30
x = 10



5. Josh graphed the function f(x) = -3(x - 1)2 + 2. He then graphed the function g(x) = -3(x - 1)2 - 5 on the same coordinate plane. The vertex of g(x) is

Answer: (1) 7 units below the vertex of f(x)
The vertex of f(x) is (1, 2). The vertex of g(x) is (1, -5).
Therefore the vertex of g(x) is 7 units lower than f(x).


6. A survey was given to 12th-grade students of West High School to determine the location for the senior class trip. The results are shown in the table below.

Niagara FallsDarien Lake New York City
Boys5674 103
Girls7192 88

To the nearest percent, what percent of the boys chose Niagara Falls?

Answer: (2) 24
Sum the number of boys by adding the three responses: 56 + 74 + 103 = 233.
Divide the number for Niagara Falls by 233, and make it a percent: 56 / 233 * 100% = 24%

Note that you don't need to use any of the data for the girls.


7. Which type of function is shown in the graph below?

Answer: (2) exponential
Basically, a definition question. Linear functions and absolute value functions are straight lines, not curves.
A square root bends the other way, with smaller increases in y and x gets larger.


8. The expression 16x2 - 81 is equivalent to

Answer: (3) (4x - 9)(4x + 9)
The Difference of Squares Rule: when you have the difference between two terms which are each perfect squares, take the square root of each term and write the conjugates. (That is, write the two terms twice, once with a plus between them, and once with a minus.)


9. The owner of a landscaping business wants to know how much time, on average, his workers spend mowing one lawn. Which is the most appropriate rate with which to calculate an answer to his question?

Answer: (4) hours per lawn
Time per lawn: time can be measured in hours and lawns in the unit measurement.
Lawns per day doesn't give the amount of time for one lawn.


10. A ball is thrown into the air from the top of a building. The height, h(t), of the ball above the ground t seconds after it is thrown can be modeled by h(t) = -16t2 + 64t + 80. How many seconds after being thrown will the ball hit the ground?

Answer: (1) 5
Put the equation in your graphing calculator, and check the table. It has zeroes at -1 and 5. Reject -1 because negative time makes no sense.

The long way:
-16t2 + 64t + 80 = 0 -- Divide by -16 to get smaller numbers.
t2 - 4t - 5 = 0
(t - 5)(t + 1) = 0
t = 5 or t = -1, reject -1


11. Which equation is equivalent to y = x2 + 24x - 18?

Answer: (1) y = (x + 12)2 - 162
You don't have to complete the square because choices are given.
Actually, you wouldn't have to complete the square even without the choices.

Put the graph into the calculator and look at the table. Where is the vertex? It is at (-12, -162).
You can also see that (-12, 126), (12, 162) and (12, 126) are not even points on the parabola.

Next method: the vertex is on the Axis of Symmetry, so x = -b / (2a) = -24/(2*1) = -12.
Eliminate choices (3) and (4).
Then y = (-12)2 + 24(-12) - 18 = -162.


12. When (x)(x - 5)(2x + 3) is expressed as a polynomial in standard form, which statement about the resulting polynomial is true?

Answer: (2) The leading coefficient is 2.
(x)(x)(2x) = 2x3, which is be the leading term, with the highest exponent. This also means that the degree is 3, and not 2.
The constant will be (-5)(3) = -15, not 2. And there will be more than these 2 terms.


13. The population of a city can be modeled by P(t) = 3810(1.0005)7t , where P(t) is the population after t years. Which function is approximately equivalent to P(t)?

Answer: (2) P(t) = 3810(1.0035)t
(1.0005)7t = (1.00057)t.
1.0005 to the 7th power is approximately 1.0035.
The rest of the equation remains unchanged.


14. The functions f(x) and g(x) are graphed on the set of axes below.


For which value of x if f(x) =/= g(x)?

Answer: (3) 3
The graphs intersect at x = -2, x = -1 and x = 2. At x = 3, g(3) = 0, but f(3) is off the top of the graph.


15. What is the range of the box plot shown below?

Answer: (1) 7
The range is 8 - 1 = 7.
The difference 6 - 4 (which is 2) would be the Interquartile Range, which is different.


16. Which expression is not equivalent to 2x2 + 10x + 12?

Answer: (3) (2x + 3)(x + 4)
Not to be a broken record, but you could put each of these into the graphing calculator and see if they give you the same graph. However, in this case, that would actually take more time.

Just looking at the choices, you should immediately see that (2) and (4) are equivalent to each other because (2x + 6) is the same as 2(x + 3). So neither of those in the answer.

Likewise, there is no way that both (1) and (3) could be equivalent: switching the 3 and 4 will result in a different middle term. So check those:

(2x + 4)(x + 3) = 2x2 + 6x + 4x + 12 = 2x2 + 10x + 12. Check!
(2x + 3)(x + 4) = 2x2 + 8x + 3x + 12 = 2x2 + 11x + 12. Wrong!


17. The quadratic functions r(x) and q(x) are given below

xr(x)
-4-12
-3-15
-2-16
-1-15
0-12
17

q(x)=x2 + 2x - 8

The function with the smallest minimum value is

Answer: (3) r(x), and the value is -16
The minimum for r(x) is -16.
Graph q(x) and find the vertex. Or use the Axis of Symmetry, x = -2/(2(1)) = -1.
Evaluate q(-1) = (-1)2 + 2(-1) - 8 = 1 - 2 - 8 = -9.
The function with the smallest minimum value is r(x), and that value is -16 (which occurs when x = -2, but that is NOT what is being asked).


18. A child is playing outside. The graph below shows the child's distance, d(t), in yards from home over a period of time, t, in seconds.

Which interval represents the child constantly moving closer to home?

Answer: (1) 0 < t < 2
If the child is moving toward home at a constant rate, the distance is decreasing steadily.
That happens between 0 and 2 seconds. at you're accidentally get the "right" answer to appear by mistake.


19. If a1 = 6 and an = 3 + 2(an - 1)2, the a2 equals

Answer: (1) 75
a2 = 3 + 2(a2 - 1)2 = 3 + 2(a1)2
= 3 + 2(6)2 = 3 + 2(36) = 3 + 72 = 75


20. The length of a rectangular patio is 7 feet more than its width, w. The area of a patio, A(w), can be represented by the function

Answer: (2) A(w) = w2 + 7w
Length = w + 7. Area = length * width = (w + 7)(w) = w2 + 7w.


21. A dolphin jumps out of the water and then back into the water. His jump could be graphed on a set of axes where x represents time and y represents distance above or below sea level. The domain for this graph is best represented using a set of

Answer: (4) positive real numbers
Forget about the above and below the water. They asked for the domain, which is the x value, which in this case represents time. Negative time makes no sense.

I don't like this problem because it leaves out 0, which is not positive, but should be part of the domain. However, the answer key tells me that it is not part of the domain.


22. Which system of linear equations has the same solution as the one shown below?


x - 4y = 10
x + y = 5

Answer: (1) 5x = 10, x + y = 5
Change x + y = 5 into y = 5 - x
That makes the first equation x - 4(5 - x) = -10
and then x - 20 + 4x = -10
So 5x - 20 = -10
And 5x = 10, which is choice (1).


23. Which interval represents the range of the function h(x) = 2x2 - 2x - 4?

Answer: (4) [4.5, ∞)
It's a parabola that opens upwards, so the upper boundary is infinity. The lower boundary is the minimum point at the vertex. Just looking at the choices, the answer should be obvious once you realize what the incorrect answer is.

You can graph this and look for the vertex.
Or use the Axis of Symmetry: x = -b/(2a) = -(-2)/(2(2)) = 0.5, which is NOT the range.
h(.5) = 2(.5)2 - 2(.5) - 4 = -4.5, which is part of the range, so use [ not (.


24. What is a common ratio of the geometric sequence whose first term is 5 and third term is 245?

Answer: (1) 7
The second term must be n/5 = 245/n, so n2 = (5)(245)
n = SQRT(1225) = 35
35 / 5 = 7, and 245 / 35 = 7
The Common ratio is 7.

End of Part I

How did you do?

Questions, comments and corrections welcome.

Wednesday, August 21, 2019

Dollars to Doughnuts

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

It's an old saying, so I went with an old spelling.

When I was a kid, $6 vs 6 donuts would have been about 3:1 odds or so. Now, it could be 1:2 or 1:3 or worse, depending on where you shop and the kind of donuts.




Come back often for more funny math and geeky comics.




Monday, August 19, 2019

Caught in a Trap

(Click on the comic if you can't see the full image.)

(C)Copyright 2019, C. Burke.

Y can't U C?
No need for suspicious minds.

Previous mentions and appearances by LVI were Agent 56 Lives! and 56 is 75.




Come back often for more funny math and geeky comics.