Wednesday, July 31, 2019

Commentary

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(C)Copyright 2019, C. Burke.

Now I remember why I don't make social commentary on social media -- I can't keep up with the math!

I'm a firm believer in not veering off either side of the road. I don't need to alienate anyone. Problem is the bigger your audience, the more you have to stick to, shall I say, ideal forms, allowing others to fill in the specifics?

First of all, Thank you to Bill R., who inadvertently provided a portion of the above dialogue. It capsulized it better than my verbosity could have. Second, the commentary portion actually succeeded because of those who commented on that, I got positive feedback from people who would not likely agree with each other. Thankfully, the examples were few, mostly esoteric, but not politically charged, which is what I feared. Likewise, it was difficult to provide anyone with an example of what I was talking about in the media today, unless I gave two examples, one for each side.

And I know doing that wouldn't have made too many people happy, and they'd have to examine themselves, or else call their own side into question. I know fewer than a dozen people on line who can do that. The rest are likely (though not definitely) to go supercomputer, supercomputer. (That is, "cray cray")

Best parts of the discussion were: the acknowledgement of the point I was making, the noting that mixture of notation didn't make mathematical sense but it got the point across, and comments that my statement could be "fixed" simply enough to give either or True or False outcome, whichever interpretation was desired. ("False", from the comments I made.)

The worst parts: people treating it like it was a Doctorate-level lecture, wanting to prove through tables and diagrams that it was Wrong Wrong Wrong Wrong Wrong, while ignoring the context, and not bothering to attempt a correction. Apparently, simplicity wasn't in their wheelhouses. If it was my four-foot swimming pool, they'd be in over their heads debating the proper construction of a breathing apparatus instead of just standing up and taking a breath of fresh air.

Am I subtweeting, subtexting, subblogging or whatever? I don't know. I don't know the individuals in question. I muted a few, and blogged one who damned me with faint praise (and probably thinks I'm too dumb to know better). But it's my blog, and this is me working it out in a nice small forum, among friends. Anyone is free to comment.

And I'm free to delete those comments as spam. If you go back to my oldest comments, you'll see that I've left up some doozies, so I don't delete lightly. Or mute for that matter. I've muted five people on Twitter EVER. Three of them this week.




Come back often for more funny math and geeky comics.




Friday, July 26, 2019

Age vs. Steps

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(C)Copyright 2019, C. Burke.

Based on empirical evidence, collected over a long period.

Originally, this comic came out of a comment that the Brooklyn Bridge seems to have gotten longer since I last crossed it. Then I realized that it used to seem shorter than when I was a kid. When you realized what shape the graph takes, the rest is self-explanatory.

Except for the roadway. I couldn't come up with a clever tag for the roadway for whose steps those might represent.




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Wednesday, July 24, 2019

Trigonometry Jones & the Ancient Hex

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(C)Copyright 2019, C. Burke.

This was originally supposed to have a "Focus!" "Directrix!" joke. That didn't happen. Probably because of how I drew the rope bridge.

Plus I knew I wanted the Trigger joke more. Oddly, the hardest part were the villains, who were originally the upside-down triangles from Temple of Duran, and then a title. Good thing I went with Hexagons, I guess!




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Saturday, July 20, 2019

New World

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(C)Copyright 2019, C. Burke.

That's one small comic for me. One giant chunk of my life spent doing this.

I knew I had to do something for Comic #1492. This idea did finally occur to me, but the timing was off -- until things In Real Life interfered and I didn't update for a week or so.

The connection between the two events isn't a new idea. Salvador Dali's painting of Columbus landing in America hints at walking on the Moon, or so the museum tour guide says.




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Thursday, July 18, 2019

Oh, What a Sine!

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(C)Copyright 2019, C. Burke.

A big hit for The Four Secants.




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Tuesday, July 16, 2019

Store Cards

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(C)Copyright 2019, C. Burke.

Art imitates Life, more often than I'd care to admit.

More updates this week. I hope.




Come back often for more funny math and geeky comics.




Sunday, July 14, 2019

June 2019 Common Core 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, triangle ABC is graphed. Triangles A'B'C' and A"B"C", the images of triangle ABC, are graphed after a sequence of rigid motions.


Identify which sequence of rigid motions maps triangle ABC onto triangle A'B'C' and then maps triangle A'B'C' onto triangle A"B"C".

Answer: (4) a reflection followed by a rotation
You can see that ABC is flipped over so that B' is the farthest point from B. It is then rotated 90 degrees clockwise to get to the final image.


2. The table below shows the population and land area, in square miles, of four counties in New York State at the turn of the century.


Which county had the greatest population density?

Answer: (3) Niagara
Population density would be the greatest number of people per square mile. So divide the number of people by the Area (square miles).

There is a shortcut. Look at Niagara. It has more people and a smaller land area than either Broome or Saratoga, so those two choices can be eliminated. You probably still want to compare it to Dutchess.

Dutchess: 280,150 / 801.59 = 349.49...
Niagara: 219,846 / 522.95 = 420.39...
Niagara has greater population density.


3. If a rectangle is continuously rotated around one of its sides, what is the three-dimensional figure formed?

Answer: (2) cylinder
A rectangle will produce a cylinder. A triangle will produce a cone. A semicircle will produce a sphere.
A rectangular prism is not possible to form by rotating a 2-D figure in 3 dimensions.


4. Which transformation carries the parallelogram below onto itself?

Answer: (4) a rotation of 180° counterclockwise about the origin
Very simply: Choices (1), (2), and (3) will all turn the horizontal lines into vertical lines, so they cannot be the answer.
Only (4) keeps the horizontal lines as well as not changing the slopes of the slanted lines.


5. After a dilation centered at the origin, the image of CD is C'D'. If the coordinates of the endpoints of these segments are C(6,-4), D(2,-8), C(9,-6), and D(3,-12), the scale factor of the dilation is

Answer: (1) 3/2
9/6 = 3/2. -12/-8 = 3/2.


6. A tent is in the shape of a right pyramid with a square floor. The square floor has side lengths of 8 feet. If the height of the tent at its center is 6 feet, what is the volume of the tent, in cubic feet?

Answer: (2) 128
V = (1/3)(6)(8)(8) = (2)(8)(8) = (2)(64) = 128.
You could've put the entire thing into a calculator, but that's how I did it in my head.


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.

Wednesday, July 10, 2019

June 2019 Common Core Geometry Regents Parts 3 and 4

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

June 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. Riley plotted A(-1,6), B(3,8), C(6,-1), and D(1,0) to form a quadrilateral. Prove that Riley’s quadrilateral ABCD is a trapezoid.
[The use of the set of axes on the next page is optional.]

Riley defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Riley’s definition to prove that ABCD is not an isosceles trapezoid.

Answer:
Graphing the quadrilateral may help, so there isn't a reason not to. You already know that it is a trapezoid, so if you make a mistake on the graph, it should be obvious enough to fix. Also, graphing it will tell you which two slopes to check.

A trapezoid has one pair of parallel sides, and parallel sides have the same slope. You can also show that the other two sides are not parallel, with different slopes, but this is not necessary.

AD: (6 - 0) / (-1 - 1) = 6/-2 = -3
BC: (-1 -8) / (6 - 3) = -9/3 = -3 AD || BC.
Riley's quadrilateral is a trapezoid because it has a pair of parallel sides.

Next, use the distance formula to find the length of the diagonals. If they are the same, then it is an isosceles trapezoid. If there are not the same length, the trapezoid is not isosceles.

AC: sqrt ( (-1 - 6)2 + (6 - -1)2) = sqrt(49 + 49) = sqrt(98)
BD: sqrt ( (8 - 0)2 + (3 - 1)2) = sqrt(64 + 4) = sqrt (68)
AC =/= BD, so the trapezoid is not isosceles.


33. A child-sized swimming pool can be modeled by a cylinder. The pool has a diameter of 6 1/2 feet and a height of 12 inches. The pool is filled with water to 2/3 of its height. Determine and state the volume of the water in the pool, to the nearest cubic foot.

One cubic foot equals 7.48 gallons of water. Determine and state, to the nearest gallon, the number of gallons of water in the pool.

Answer:
Half of 6.5 feet diameter is 3.25 feet radius. Twelve inches is 1 foot, and 2/3 of that is 2/3 feet.
V = pi * r2 * h = (3.141592)(3.25)2(2/3) = 22.122
The pool holds about 22 cubic feet of water

22 * 7.48 = 164.56 = 165 gallons of water.


34. 4 Nick wanted to determine the length of one blade of the windmill pictured below. He stood at a point on the ground 440 feet from the windmill’s base. Using surveyor’s tools, Nick measured the angle between the ground and the highest point reached by the top blade and found it was 38.8°. He also measured the angle between the ground and the lowest point of the top blade, and found it was 30°.


Determine and state a blade’s length, x, to the nearest foot.

Answer:
There are two right triangles. You need to find the opposite side (the height) of each of them. The difference between the two is the height of the blade, x.

You know the adjacent, 440, and you are looking for the opposite, so you need to use tan for both.

tan (30) = y / 440
y = 440 * tan (30) = 254.034...

tan (38.8) = z / 440
y = 440 * tan (38.8) = 353.769.--

353.8 - 254.0 = 99.8 = 100 feet.


Part IV

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

35. Quadrilateral MATH, HM = AT, HT = AM, HE is perpendicular to MEA, and HA is perpendicular to AT

Prove TA * HA = HE * TH

Answer:
I don't remember another time that they asked you to prove something like this.
Work backward for a step here.
If TA * HA = HE * TH then TA / TH = HE / HA.
In other words, sides are proportional. So you can show that two triangles are proportional, which will get you here. Two triangles will have proportional sides if they are similar, and similar triangles have three pairs of congruent angles, but you only have to find two.

StatementsReasons
1. Quadrilateral MATH, HM = AT, HT = AM, HE is perpendicular to MEA, and HA is perpendicular to AT1. Given
2. Angle HEA and angle TAH are right angles.2. Perpendicular lines form right angles
3. Angle HEA = Angle TAH3. All right angles are congruent.
4. MATH is a parallelogram4. A quadrilateral with two pairs of congruent sides is a parallelogram.
5. MA || TH5. Opposite sides of a parallelogram are parallel. (Definition of parallelogram)
6. Angle THA = Angle EAH6. Alternate interior angles.
7. Triangle HEA ~ Triangle TAH7. AA Theorem
8. TA / TH = HE / HA 8. Corresponding sides of similar triangles are proportional.
9. TA * HA = HE * TH 9. In a proportion, the cross-products are equal.

End of Exam

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

Tuesday, July 09, 2019

Ken-Do #15: Sin-Cos-Tan

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

(C)Copyright 2019, C. Burke.

What web of puns is he spinning now?
The bottom text was supposed to be the after text, but I didn't have a bottom text. Oh well.

I haven't done a lot of Ken-Do comics because I still have to go back and re-do the originals.




Come back often for more funny math and geeky comics.




Sunday, July 07, 2019

June 2019 Common Core Geometry Regents Part 2 (open ended)

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 II

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


25. Triangle A'B'C' is the image of triangle ABC after a dilation with a scale factor of 1/2 and centered at point A. Is triangle ABC congruent to triangle A'B'C'? Explain your answer.

Answer:
Triangle ABC is not congruent to triangle A'B'C' because a dilation of scale factor 1/2 does not preserve size.

It doesn't ask, but ABC would be similar to A'B'C' because shape is preserved, but size is not.


26. Determine and state the area of triangle PQR, whose vertices have coordinates P(-2,-5), Q(3,5), and R(6,1).
[The use of the set of axes below is optional.]

Answer:
To find the area of the triangle, you need a base and an altitude, which are at right angles to each other. This is trivial if the triangle is a right triangle and two sides are perpendicular.

If you graph it, if looks like angle R could be a right angle, so check the slopes of QR and RP.
QR: (5 - 1) / (3 - 6) = 4 / -3 = -4/3
RP: (1 - -5) / 6 - -2) = 6 / 8 = 3/4
The slopes are inverse reciprocals so they are perpendicular, meaning they can be used as the base and altitude.

Next, find the length of each
QR: SQRT( (3-6)2 + (5-1)2) = SQRT ( (-3)2 + (4)2)) = Sqrt(9+16) = sqrt(25) = 5
QR: SQRT( (6 - -2)2 + (1 - -5)2) = SQRT ( (8)2 + (6)2)) = Sqrt(64+36) = sqrt(100) = 10

A = 1/2(5)(10) = 25

Another method that doesn't rely on slopes and distance formula:
Graph the triangle.
Make a rectangle with P at a vertex and Q and R as points on the sides of the rectangle.
If you count boxes, you will find that the rectangle has an area of (8)(10) = 80.
There are four triangles within the rectangle, three of which are right triangles which line up with the grid. You can find the areas of these very easily: one is 10 by 5, one is 3 by 4, and the last is 6 by 8.
Those 3 have areas of 1/2(10)(5) = 25, 1/2(3)(4) = 6, 1/2(6)(8) = 24.
80 - (25 + 6 + 24 ) = 80 - 55 = 25.
This is a valid approach to take using the provided set of axes.


27. A support wire reaches from the top of a pole to a clamp on the ground. The pole is perpendicular to the level ground and the clamp is 10 feet from the base of the pole. The support wire makes a 68° angle with the ground. Find the length of the support wire to the nearest foot.

Answer:
The angle is on the ground. The distance from the clamp to the pole is the side adjacent to the pole. The length of the wire is the hypotenuse. That means using cosine.

Cos 68° = 10 / x
x (cos 68°) = 10
x = 10 / (cos 68°) =26.694...
The wire is approximately 27 feet.


28. In the diagram below, circle O has a radius of 10.

If mAB = 72°, find the area of shaded sector AOB, in terms of π.

Answer:
Area of a circle is πr2.
Area of a sector of circle is (central angle/360) πr2.
A = (72/360) π (10)2
A = 20 π

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. On the set of axes below, triangle ABC = triangle STU.

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

Answer:
I don't like this question because students will second-guess themselves because of its simplicity. It says "a sequence of rigid motions" when, in fact, it can be done with one.

You can map ABC onto STU with a rotation of 90 degrees counterclockwise about the origin.

You might have done is about point B, for example, and then followed that with a translation of T2,-6.


30.
In right triangle PRT, m∠P = 90°, altitude PQ s drawn to hypotenuse RT, RT = 17, and PR = 15.


Determine and state, to the nearest tenth, the length of RQ.

Answer:
RQ is the long leg of right triangle PQR. PR is the hypotenuse of PQR, and the long leg of PRT. RT is the hypotenuse of PRT.
Because PRT is similar to PQR, you can write a proportion:
(RQ / PR) = (PR / RT)
(RQ / 15) = (15 / 17)
17 RQ = (15)(15)
17 RQ = 225
RQ = 225 / 17 = 13.23...
RQ is approximate 13.2.


31. Given circle O with radius OA, use a compass and straightedge to construct an equilateral triangle inscribed in circle O.
[Leave all construction marks.]

Answer:
They have asked a very similar question before. That time it was for an inscribed hexagon, which involves many of the same steps. There are only so many construction questions that they can ask.

Say we want to make equilateral triangle ABC within circle O. That means that angle BAC will be an inscribed angle measuring 60 degrees and intercepting an arc BC with a measure of 120 degrees. The same for the other two angles.

We want the three points to be one third of the triangle apart from each other. And this is where the radius comes in. If I had a protractor (don't do this, I'm just explaining), I could draw the diameter. I could then split each semicircle in three by creating 60 degrees angles. Three 60-degree angles add up to 180 degrees. And the radii are all the same size. You could then make a hexagon by connecting the endpoints of those six radii. It would be a regular hexagon, as all the sides are the same length, and, in fact, the same length as each radius because you would have six little equilateral triangles.

Where am I going with all this? Simple:

Use the length of the radius to create six arc around the circle, starting and ending with point A. Each arc will measure 60 degrees. Mark the second one B and the fourth one C.

With the straightedge, draw AB, BC and CA. You're done.

Yes, I could have just told you to do that, but where would the fun in that be? Wouldn't you want to know why you could do it this way?

(The above image was done on a computer so it's approximate.)

End of Part II

How did you do?

Questions, comments and corrections welcome.

Saturday, July 06, 2019

June 2019 Algebra I Regents Part 1

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

Omitted images will be added soon.

January 2019 Algebra I, Part I

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


1. The expression w4 - 36 is equivalent to

Answer: (4) (w2 + 6)(w2 - 6)
Difference of Squares rule. You get the conjugates consisting of the square root of each term. ("Conjugates" means the same two terms, but one binomial has a plus and the other a minus.)
When multiplied the two w2 terms with negate each other. (Cancel out.)


2. If f(x) = 4x + 5, what is the value of f(-3)?

Answer: (2) -7
Substitute and evaluate
f(-3) = 4(-3) + 5 = -12 + 5 = -7


3. Which relation is not a function?

Answer: (4)
The map in (4) shows two arrows coming from 2 in the domain to two different elements of the range. This is not allowed in a function.
In Choice (1), no x value is repeated.
In Choice (2), the graph passes the vertical line test.
In Choice (3), you have a linear equation in standard form. All linear equations, with the exception of vertical lines (such as x = 3), are functions.


4. Given: f(x) = (x - 2)2 + 4 and g(x) = (x - 5)2 + 4
When compared to the graph of f(x), the graph of g(x) is

Answer: (2) shifted 3 units to the right
The vertex of f(x) is (2, 4).
The vertex of g(x) is (5, 4).
To get to g(x) from f(x), you have to shift 3 units to the right.


5. Students were asked to write 6x5 + Bx - 3x3 + 7x7 in standard form. Shown below are four student responses.


Anne: 7x7 + 6x5 - 3x3 +Bx
Bob: -3x3 + 6x5 + 7x7 +Bx
Carrie: Bx + 7x7 + 6x5 - 3x3
Dylan: Bx - 3x3 + 6x5 + 7x7
Which student is correct?

Answer: (1) Anne
Standard form puts the term with the highest exponent first. Second highest exponent second, etc.
Anne is the only one to put the x7 term first.


6. The function f is shown in the table below

xf(x)
01
13
29
327

Which type of function best models the given data?

Answer: (1) exponential growth function
Quick elimination: there is not a constant rate of change, so it is not linear. So (3) and (4) are incorrect. The values are increasing, so it is not decay, and (2) is incorrect.
The f(x) values are increasing and have a common ratio, 3/1 = 9/3 = 27/9. It is an example of exponential growth.
More specifically, it is f(x) = 3x


7. Which expression results in a rational number?

Answer: (1) Sqrt(2) * Sqrt(18)
The sum of two irrational numbers will be irrational. The product of a rational number and an irrational number will be irrational.
The product of two irrational number could be rational, if, in this instance, it produces a perfect square under the radical sign. The square root of 36 is 6, which is rational.


8. A polynomial function is graphed below.


Which function could represent this graph?

Answer: (3) f(x) = (x - l)(x2 - 4)
The zeroes of the function are -2, 1, and 2.
The factors of the function must be (x + 2), (x - 1) and (x - 2).
The product of (x + 2)(x - 2) is (x2 - 4).

Looking at the choices, you know that there are three real roots.
Neither (x2 + 2) nore (x2 + 4) have real roots. They wouldn't touch the x-axis.
Choice (2) is no good because (x2 - 2) would have roots at the square root of 2, which would be in the middle of 1 and 2 (and -1 and -2) on the graph.


9. When solving p2 + 5 = Sp - 7, Kate wrote p2 + 12 = Sp. The property she used is

Answer: (4) the addition property of equality
She added 7 to both sides of the equation. Associative property changes the groupings, which didn't happen. Commutative property changes the order, which didn't happen. The Distributive Property involves parentheses, which aren't there.


10. David wanted to go on an amusement park ride. A sign posted at the entrance read "You must be greater than 42 inches tall and no more than 57 inches tall for this ride." Which inequality would model the height, x, required for this amusement park ride?

Answer: (1) 42 < x < 57
The variable has be in between 42 and 57. Only choice number (1) makes any sense.
Choice (2) literally says 42 > 57. Not true.
Choice (3) would work if it said AND instead of OR because, for example, 10 is less than 57, so that makes the entire condition true.
Choice (4) is almost, but not quite, the conditions for people cannot use the ride.


11. Which situation can be modeled by a linear function?

Answer: (3) An amusement park allows 50 people to enter every 30 minutes.
Every 30 minutes 50 people enter is a constant rate, so it is a linear function.
Choice (1) is exponential growth.
Choices (2) and (4) are exponential decay.


12. Jenna took a survey of her senior class to see whether they preferred pizza or burgers. The results are summarized in the table below.

PizzaBurgers
Male 23 42
Female31 26
Of the people who preferred burgers, approximately what percentage were female?

Answer: (2) 38.2
26 / (26 + 42) = 0.38235...

Just looking at the numbers, you could tell that it's a little more than a third.
Only one choice is a little more than a third. Another is less than a third, and the next is almost one-half.


13. 2a - 8b is solved for a, the result is

Answer: (4) a > ­-l5b
3a + 7b > 2a - 8b
a + 7b > - 8b
a > -15b


14. Three functions are shown below:


A: g(x) = -3/2 x + 4
B: f(x) = (x + 2)(x + 6)
C: (graph)

Which statement is true?

Answer: (3) B has a minimum and C has a maximum.
B is a parabola that opens upward, so it has a minimum. The graph of C shows a maximum.
Choice (1) is incorrect because the zeroes of B are -2 and -6, but C shows -2 and -4.
Choice (2) is incorrect because the y-intercept of A is 4, but the y-intercept of B is (2)(6) = 12.
Choice (4) is incorrect because A has no minimum -- it is a linear function.


15. Nicci’s sister is 7 years less than twice Nicci’s age, a. The sum of Nicci’s age and her sister’s age is 41. Which equation represents this relationship?

Answer: (2) a + (2a - 7) = 41
7 less than twice a is 7 less than 2a, which is 2a - 7. It is NOT 7 - 2a.
(Consider: if I have 7 less than 100, I have 100 - 7 = 93, not 7 - 100 = -93.)
The sum of that expression and a is 41, so a + (2a - 7) = 41.


16. The population of a small town over four years is recorded in the chart below, where 2013 is represented by x 0. [Population is rounded to the nearest person]

Year 2013 2014 2015 2016
Population 3810 3943 4081 4224

The population, P(x), for these years can be modeled by the function P(x) = abx, where b is rounded to the nearest thousandth. Which statements about this function are true?

I. a = 3810
II. a = 4224
III. b = 0.035
IV. b = 1.035

Answer: (2) I and IV
Put the numbers in Lists and do an Exp regression. You'll get a = 3810 and b = 1.035.

WITHOUT PUTTING IT IN THE CALCULATOR: The first number is 3810 and the last is 4224. The base has to be 3810. This is an example of exponential growth, not decay, so the base has to be greater than 1.


17. When written in factored form, 4w2 - 11w + 3 is equivalent to

Answer: (3) (4w + 1)(w - 3)
Looking quickly, all four will give you 4w2 and -3. So you only have to concern yourself with the inner and outer terms.
(4w)(-3) = -12w and (1)(w) = w, -12w + w = -11w, which is what we want.


18. Which ordered pair does not represent a point on the graph of y = 3x2 - x + 7?

Answer: (3) (1.25, 10.25)
If you plot this, you can manipulate the table of values to show you multiples of .25. Or you can use the Trace function to zero in on the points. You can also use a list of a set to check all four values at once.

Or just check one at a time:
3(-1.5)^2 - (-1.5) + 7 = 15.25
3(0.5)^2 - (0.5) + 7 = 7.25
3(1.25)^2 - (1.25) + 7 = 10.4375
3(2.5)^2 - (2.5) + 7 = 23.25

If you brute force it, don't stop at the first wrong answer. Keep going. If you get two wrong answers, you know you mis-entered something. Odds are low that you're accidentally get the "right" answer to appear by mistake.


19. Given the following three sequences:

I. 2, 4, 6, 8, 10...
II. 2, 4, 8, 16, 32...
III. a, a + 2, a + 4, a + 6, a + 8...
Which ones are arithmetic sequences?

Answer: (2) I and III, only
Each term in I increases by 2. Each term in II doubles the previous, which is geometric, not arithmetic. Each term in III is 2 more than the one before it.


20. A grocery store sells packages of beef. The function C(w) represents the cost, in dollars, of a package of beef weighing w pounds. The most appropriate domain for this function would be

Answer: (4) positive rational numbers
Weight of beef does not have to be in whole number pounds, so integers is not appropriate.
Likewise, the weight have to be a non-negative number. While zero is a valid number for weight, it doesn't make sense for a sale -- you couldn't sell nothing. So positive rational numbers is the best.


21. The roots of x2 - 5x - 4 = 0 are

Answer: (2) (5 + SQRT(41) ) / 2
If either Choice (1) or (3) were correct, the final term would have to be + 4, not - 4.
(x - 1)(x - 4) and (x + 1)(x + 4) both have + 4 as the final term.

The quadratic formula (as the Axis of Symmetry formula) starts with -b.
A -(-5) is +5, which means the correct answer is (2) not (4).


22. The following table shows the heights, in inches, of the players on the opening-night roster of the 2015-2016 New York Knicks.

84 80 87 75 77 79 80 74 76 80 80 82 82
The population standard deviation of these data is approximately

Answer: (1) 3.5
No calculations necessary. Okay, one small one.
The range of the data is 87 - 74 = 13.
The standard deviation is much smaller than that, and the only possible choice is (1).
Without doing any other work, I would estimate that 79.7 is the mean and 80, which is obviously the mode, is likely the median as well.


23. W A population of bacteria can be modeled by the function f(t) = 1000(0.98)t, where t represents the time since the population started decaying, and f(t) represents the population of the remaining bacteria at time t. What is the rate of decay for this population?

Answer: (2) 2%
Two percent decay means that 98% of the bacteria remain, and that is written as 0.98.


24. Bamboo plants can grow 91 centimeters per day. What is the approximate growth of the plant, in inches per hour?

Answer: (1) 1.49
You need to convert centimeters per day into inches per hour.
The reference sheet gives you 1 inch 2.54 centimeters, and, of course, there are 24 hours in one day.

So (91 cm / day) * (1 day / 24 hrs) * (1 in / 2.54 cm) = 1.49278...

End of Part I

How did you do?

Questions, comments and corrections welcome.

Thursday, July 04, 2019

The Singers of the Declaration of Independence

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

(C)Copyright 2019, C. Burke.

Bet you thought it was a typo. Okay, it's not as catchy as the Preamble song, but what is?

John Dickinson, who originally wrote "The Liberty Song" in 1768, did not sign the Declaration, but contrary to popular musicals, he didn't leave Congress either. Also Judge James Wilson of Pennsylvania was not Dickinson's shadow, but it played well on stage.

Final note: when I realized how large I needed to make the comic, I expanded the field to a size of 1776 by 704. Because I could, and no one could tell me otherwise. (Except maybe a publisher some day.)




Come back often for more funny math and geeky comics.




Wednesday, July 03, 2019

June 2019 Algebra I Regents Parts 3 & 4

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

June 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. A school plans to have a fundraiser before basketball games selling shirts with their school logo. The school contacted two companies to find out how much it would cost to have the shirts made. Company A charges a $50 set-up fee and $5 per shirt. Company B charges a $25 set-up fee and $6 per shirt.

Write an equation for Company A that could be used to determine the total cost, A, when x shirts are ordered. Write a second equation for Company B that could be used to determine the total cost, B, when x shirts are ordered.

Determine algebraically and state the minimum number of shirts that must be ordered for it to be cheaper to use Company A.

Answer:
Write equations from each verbal expression.
A = 5x + 50
B = 6x + 25
To find when A is less than B, start with A < B and then substitute and solve.


A < B
5x + 50 < 6x + 25
-x < -25
x > 25

The number of shirts must be more than 25, so the minimum number of shirts is 26.




34. Graph y = f(x) and y = g(x) on the set of axes below.

f(x) = 2x2 - 8x + 3
g(x) = -2x + 3

Determine and state all values of x for which f(x) = g(x).

Answer:
A parabola and a line will intersect at either one or two points. If it's two, you have to make sure you have them both on your graph.
Label which line is which, even if it's "obvious".
Label the points of intersection.
Answer the question, supplying only the values of x (not f(x) or g(x)).

Check the graph:

The values of x for which f(x) = g(x) are 0 and 3.
Do NOT write (0, 3), which looks like a point on the graph, or an interval. Neither would be a correct answer.


35. The table below shows the number of hours ten students spent studying for a test and their scores.

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

State the correlation coefficient of this line, to the nearest hundredth.

Explain what the correlation coefficient suggests in the context of the problem.

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 = 7.79, b = 34.27 and r = .98, rounded to the nearest hundredth.

The linear regression is y = 7.79x + 34.27

The correlation coefficient is r = 0.98

In the context of this problem, the correlation coefficient suggest a strong positive correlation between the number of hours spent studying and test scores.
(It's not enough to just say that it's a strong positive correlation.)


36. A system of inequalities is graphed on the set of axes below.

State the system of inequalities represented by the graph.

State what region A represents.

Answer:
Working backward, you need to find the slope and y-intercept of each of the lines.
Note that the line with the negative slope is broken, so it is not equal. Also, note that both lines are shaded below, so you need to use less than, or less than or equal to.

The broken line has a slope of -3 and a y-intercept of 3, so
y < -3x + 3
The solid line has a slop of 2 and a y-intercept of -2, so
y < 2x - 2

Region A represents the solution to the system of inequalities.

The entire gray region represents the values of y that are less than -2x - 2 only.

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. When visiting friends in a state that has no sales tax, two families went to a fast-food restaurant for lunch. The Browns bought 4 cheeseburgers and 3 medium fries for $16.53. The Greens bought 5 cheeseburgers and 4 medium fries for $21.11.

Using c for the cost of a cheeseburger and f for the cost of medium fries, write a system of equations that models this situation.

The Greens said that since their bill was $21.11, each cheeseburger must cost $2.49 and each order of medium fries must cost $2.87 each. Are they correct? Justify your answer.

Using your equations, algebraically determine both the cost of one cheeseburger and the cost of one order of medium fries.

Answer:
Browns: 4c + 3f = 16.53
Greens: 5c + 4f = 21.11

Check by putting it into each equation. Stop if one if incorrect.
4(2.49) + 3(2.87) = 16.53?
18.57 =/= 16.53
The Greens are incorrect.

Notice that they ask you to find c and to find f. They don't ask you to find (c + f), which would have be trivial -- just subtract the two equations.

Now that I wrote that, I want to try something that I normally wouldn't do. It's a little extra work, but it uses smaller numbers and I like smaller numbers.


5c + 4f = 21.11
4c + 3f = 16.53
c + f = 4.58
(3)(c + f) = (4.58)(3)
3c + 3f = 13.74

4c + 3f = 16.53
3c + 3f = 13.74
c = 2.79
2.79 + f = 4.58
f = 1.79

By subtracting, I found the cost of 1c + 1f, which I could then multiply by three to solve the system of equations. Normally, I wouldn't do it that way, but it was something I noticed -- Number Sense to the rescue! And that saved me from doing it the following way, by finding Least Common Multiples:


4c + 3f = 16.53
5c + 4f = 21.11
(4)(4c + 3f = 16.53)
(3)(5c + 4f = 21.11)
16c + 12f = 66.12
15c + 12f = 63.33
c = 2.79

4(2.79) + 3f = 16.53
3f = 5.37
f = 1.79




End of Exam

How did you do?

Questions, comments and corrections welcome.

How Long Does It Take to Fill ...?

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

(C)Copyright 2019, C. Burke.

Oddly enough, I'm a bit late with filling the pool this year, so this comic isn't actually late for me.

However, these days, I do set a reminder timer any time I think I need to "top off" the pool.




Come back often for more funny math and geeky comics.




Tuesday, July 02, 2019

June 2019 Algebra 1 Regents, Part 2

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

June 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. Solve algebraically for x: -2/3 (x + 12) + 2/3 x = -5/4 x + 2

Answer:
You can do a few things here. You can multiply each term in the equation by 12 (3 times 4) to get rid of the denominators, or you can combine the terms on the left and deal with the fractions later.

First method


-2/3 (x + 12) + 2/3 x = -5/4 x + 2
(12)(-2/3 (x + 12)) + (12)(2/3 x) = (12)(-5/4 x) + (12)(2)
-8 (x + 12) + 8x = -15x + 24
-8x - 96 + 8x = -15x + 24
-96 = -15x + 24
-110 = - 15x
8 = x

Second method


-2/3 (x + 12) + 2/3 x = -5/4 x + 2
-2/3x - 8 + 2/3 x = -5/4 x + 2
-8 = -5/4 x + 2
-10 = -5/4 x
-40 = -5 x
8 = x



26. If C = G - 3F, find the trinomial that represents C when F = 2x2 + 6x - 5 and G = 3x2 + 4.

Answer: Multiply F by 3 and subtract it from G.
3F = 3(2x2 + 6x - 5) = 6x2 + 18x - 15

(3x2 + 4) - (6x2 + 18x - 15) = -3x2 - 18x + 19

My guess is one point for multiplying 3F and the other for combining them properly.
One computational error costs a point. Two means no credit.


27. Graph the following piecewise function on the set of axes below.

f(x) = { |x|, -5 < x < 2
{ -2x + 10 , 2 < x < 8

Answer: Important: you are given a specific range. Do NOT add arrows to the endpoints.
Also, remember at x = 2, you need an open circle to mark the end of the absolute value graph, and a closed circle to start the linear function.




28. Solve 5x2 = 180 algebraically.

Answer:

5x2 = 180
x2 = 36
x = SQRT(36)
x = +6

If you forget the -6, you lost a point.
If you divided incorrectly and didn't get a perfect square, you could have salvaged a point if you left your answer in radical form with a + symbol in front of it. However, if you estimated the radical, you probably lost the other point as well.


29. A blizzard occurred on the East Coast during January, 2016. Snowfall totals from the storm were recorded for Washington, D.C. and are shown in the table below.

Washington, D.C.
Time Snow (inches)
1 a.m. 1
3 a.m 5
6 a.m 11
12 noon33
3 p.m 36

Which interval, 1 a.m. to 12 noon or 6 a.m. to 3 p.m., has the greatest rate of snowfall, in inches per hour? Justify your answer.

Answer:
Find the rate of change for both intervals:
1 am to 12 noon: (33 - 1) / (12 - 1) = 32 / 11 = 2.909...
6 am to 3 pm (1500 in military time): (36 - 11) / (15 - 6) = 25 / 9 = 2.777...

1 am to 12 noon had the greater rate of snowfall in inches per hour.

If you made one mathematical error, you could still get a point for the final answer based on your work.


30. The formula for the volume of a cone is V = (1/3)πr2h. Solve the equation for h in terms of V, r, and π.

Answer:
This should be a "gimme" because it's one of the most common examples used in Literal Equations exercises. I'm surprised that they didn't use r so you'd have to deal with a square root!

Isolate the h by dividing by the other terms. (Dividing by 1/3 is the same as multiplying by 3.)


V = (1/3)πr2h
3V = πr2h
3V / (πr2) = (πr2h) / (πr2)
3V / (πr2) = h



31. Given the recursive formula:

a1 = 3
an = 2(an - 1 + 1)

State the values of a2, a3, and a4 for the given recursive formula.

Answer:
Just plug the numbers into the formula to come up with the next number. A consistent error will not be penalized multiple times.

a1 = 3
a2 = 2(a1 + 1) = 2(3 + 1) = 2(4) = 8
a3 = 2(a2 + 1) = 2(8 + 1) = 2(9) = 18
a4 = 2(a3 + 1) = 2(18 + 1) = 2(19) = 38


32. Determine and state the vertex of f(x) = x2 - 2x - 8 using the method of completing the square.

Answer:
Okay, I'll be the first to admit that this is an annoying question. If I wanted to know the vertex, I would use the Axis of Symmetry to find it.
If they want you to "Complete the Square", they should just say that.
It's a two point question, so you will lose a point if you use any other method to find the vertex. You will also get one point for stating the vertex without any work at all. As much as I hate to say this, if you aren't completing the square then your work doesn't matter for this problem.

Look at the middle term: -2x. Half of -2 is -1, so the square you need to complete is (x - 1)2.
If you square that binomial, the constant will be + 1, not - 8.
I'll do this is extra steps, but you can do it in fewer.


f(x) = x2 - 2x - 8
f(x) + 1 = x2 - 2x + 1 - 8
f(x) = (x2 - 2x + 1) - 8 - 1
f(x) = (x - 1)2 - 9

The vertex is (1, -9). You can check this on your calculator.

End of Part II

How did you do?

Questions, comments and corrections welcome.