Friday, July 21, 2017

Cosmic Microwaves

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(C)Copyright 2017, C. Burke.

What it says on the Tin

If the Microwave was the greatest invention since the invention of popcorn, the Popcorn button was the greatest invention since the microwave!




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Wednesday, July 19, 2017

With A Song in My Math

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(C)Copyright 2017, C. Burke.

I could go on ALL DAY!!!! ... but I won't

Alphabetically speaking, you're OK. No, you're Buddy Kaye!




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Monday, July 17, 2017

The Doctor is Still In

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(C)Copyright 2017, C. Burke.

When is Who on?

Joke's on Michele: she'll have to sit through a whole Christmas special with two old male Doctors before we get a quick introduction to the new Who.

Oh, and "Spoiler Alert!" if you somehow missed the news.




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Thursday, July 13, 2017

Just What the Teacher Inscribed!

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(C)Copyright 2017, C. Burke.

I would've put 'Mom and Dad', but I had space limitations. Space: the final frontier.

Plus, from real life, there's a good choice that it's a single parent carrying on the good traditions while the other is poisoning their minds with 'weird stuff'.




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Tuesday, July 11, 2017

Don't Think, So?

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(C)Copyright 2017, C. Burke.

I've heard variations enough to think we put Descartes in front of a dead horse, beaten with the same puns.

Of course, if we set the Wayback Machine to 2010, we've tackled Descartes before.




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Sunday, July 09, 2017

(x, why?) Mini: Think Proof

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(C)Copyright 2017, C. Burke.

Not to be confused with 'Thinkproof' or even 'Thoughtproof', which almost seems to describe some average adults I know.




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Thursday, July 06, 2017

Like, Totally Independent

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(C)Copyright 2017, C. Burke.

I really hope this reads like it did in my head.

Math joke: I can imagine an 'x' separated from the other letters saying he was 'totally independent' from the other variables.
Maybe that can be a future joke(?).




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Wednesday, July 05, 2017

June 2017: Common Core Geometry Regents, Part 1

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

Answers to Part II can be found here.

Answers to Part III can be found here.

Answers to Part IV can be found here.

June 2017, Geometry (Common Core), Part I

1.In the diagram below, triangle ABC = triangle DEF. (image omitted)
Which sequence of transformations maps triangle ABC onto triangle DEF?

(2) a reflection over the y-axis followed by a translation. ABC is flipped over the y-axis, so it is facing the opposite direction. Then it sinks down below the x-axis through a translation.


2.On the set of axes below, the vertices of triangle PQR have coordinates P(-6,7), Q(2,l), and R(-1, -3). (image omitted)
What is the area of triangle PQR?

(3) 25. If you notice, PQ is perpendicular to QR. The slope of PQ is -3/4, and the slope of QR is 4/3. Pythagorean theorem -- as well as knowledge of Pythagorean Triples -- will tell you that PQ = 10 and QR = 5. The area of a triangle is (1/2)(b)(h) = (1/2)(5)(10) = 25.
(You might also remember for several of my comics that the area of a triangle could be written as "one-half a b".)


3.In right triangle ABC, m<C = 90°. If cos B = 5/13, which function also equals 5/13?

(3) sin A. If C is the right angle, then A and B are the acute angles, which are complimentary. The sine of one acute angle will always equal the cosine of the other acute angle, but what is opposite one angle is adjacent to the other.


4.In the diagram below, m<ABC = 268°. (image omitted)
What is the number of degrees in the measure of <ABC?

(4) 46°. The complete circle is 360°. If major arc is ABC = 268, then AC = 360 - 268 = 92. The inscribed angle is half the size of the central angle, so 92 / 2 = 46°.


5.Given triangle MRO shown below, with trapezoid PTRO, MR = 9, MP = 2, and PO = 4. (image omitted)
What is the length of TR?

(4) 6. PO / MO = TR / MR. MO = 6, so 4 / 6 = TR / 9. Cross-multiply and get TR = 6.


6.A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true?

(3) The line segments are parallel, and the image is twice the length of the given line segment. The image will also be twice as far away from the center than the original line was.


7.Which figure always has exactly four lines of reflection that map the figure onto itself?

(1) Square. Lines through the midpoints of each pair of parallel sides and the two diagonals. An octagon has eight lines: through the four pairs of parallel lines and lines through the opposites vertices.


8.In the diagram below of circle 0, chord D F bisects chord BC at E. (image omitted)
If BC = 12 and FE is 5 more than DE, then FE is

(2) 9. BC is bisected, so BE = CE = 6. The rule for intersecting chords tells us that

(BE)(CE) = (DE)(FE).
(6)(6) = (x)(x + 5)
36 = x2 + 5x

Stop here. If you did this, you're doing too much work for a multiple choice question.

What are the factors of 36?
1, 36; 2, 18; 3, 12; 4, 9; 6, 6.
Which two have a difference of 5? 4 and 9, both of which are choices.
FE is the longer one (it's "5 more"), so the answer is 9.


9.Kelly is completing a proof based on the figure below. (image omitted)
She was given that <A = <EDF, and has already proven AB = DE. Which pair of corresponding parts and triangle congruency method would not prove triangle ABC = triangle DEF?

(2) BC = EF and SAS. SAS requires that the congruent angle be included between the two pairs of congruent sides. BC and EF would give you SSA instead, and that is not allowed (except in the special case of right triangles, where it is HL).


10. In the diagram below, DE divides AB and AC proportionally, m<C = 26°, m<A = 82°, and DF bisects LBDE. (image omitted)
The measure of angle DFB is

(2) 54°. DE divides the two sides proportionally, so DE is parallel to side BC. Angle AED = 26 because it is a corresponding angle. Using the Exterior Angle theorem, we know that <EDB = 82 + 26 = 108. (If you forgot that theorem, you could have found <ADE because a triangle has 180 degrees and then found DF bisects <BDE, so <BDF = <EDF = 54. Angle DFB is an alternate interior angle to EDF, so it equals 54°.

You also could have solved it by finding the angles of triangle DBF. Angle B = 180 - 82 - 26 = 72. So <DFB = 180 - 54 - 72 = 54


11.Which set of statements would describe a parallelogram that can always be classified as a rhombus?
I. Diagonals are perpendicular bisectors of each other.
II. Diagonals bisect the angles from which they are drawn.
III. Diagonals form four congruent isosceles right triangles.

(2) I and III. The diagonals of a rhombus are NOT angle bisectors, unless it's a square.
Note: There is some important information in this question that could have helped you with the proof in Part IV!


12.The equation of a circle is x2 + y2 - 12y + 20 = 0. What are the coordinates of the center and the length of the radius of the circle?

(2) A = 1000(1 + 0.013)2. Interest increases your value, so choices (1) and (3) are right out. The percent 1.3% must be converted to a decimal, which is 0.013. The correct answer is (2). You need to complete the square. Half of -12 is -6, so we need to get to (y - 6)2:

x2 + y2 - 12y + 20 = 0
x2 + y2 - 12y + 36 + 20 = 36
x2 + y2 - 12y + 36 = 16
x2 + (y - 6)2 = 42
This make the center (0, 6) and the radius 4.


13. In the diagram of triangle RST below, m<T = 90°, RS = 65, and ST = 60. (image omitted) What is the measure of <S, to the nearest degree?

(1) 23°. First of all, I hope you realized that RT is the shortest side of the triangle (not because of looks, but because of Pythagorean Triples), so choices (3) and (4) are too big.
We have the adjacent side and the hypotenuse so we need to use cosine to find the angle.
cos S = 60 / 65
S = cos-1 (60 / 65) = 22.62..., which rounds to 23.


14. Triangle A'B'C' is the image of triangle ABC after a dilation followed by a translation. Which statement(s) would always be true with respect to this sequence of transformations?
I. Triangle ABC = triangle A'B'C'
II. Triangle ABC ~ triangle A'B'C'
III. AB || A'B'
IV. AA'= BB'

As a result of discrepancies in the wording, Questions 14 does not have one clear and correct answer.
Either (1) II, only or (3) II and III were accepted.
I don't know what the discrepancy might be that would make someone not think that AB was not parallel to A'B', unless there were printing errors in some books that left out one of the accent marks.

A dilation will preserve the shape of the original object, so the original and the image must be similar.
The center point is not given, but regardless, the orientation will not change, so the slopes of the sides will not change, so the sides will be parallel.
(3) is the best answer, unless some books have a typo of some kind.


15. Line segment RW has endpoints R(-4,5) and W(6,20). Point P is on RW such that RP:PW is 2:3. What are the coordinates of point P?

(2) (0,11). Add 2 + 3 = 5, so RP is 2/5 the length of RW. Find difference of the x-values and y-values of RW and multiply by 2/5.
2/5(6 - (-4)) = 2/5(10) = 4
2/5(20 - 5) = 2/5(15) = 6
Add +4,+6 to point R(-4, 6). P(-4 + 4, 5 + 6) = P(0, 11).


16. The pyramid shown below has a square base, a height of 7, and a volume of 84. (image omitted)
What is the length of the side of the base?

(1) 6. Volume = (1/3)(Area of Base)(height)
84 = (1/3)(B)(7)
252 = 7B
36 = Area of the Base.
The base is a square, then the length of one side is the square root of 36, which is 6


17.In the diagram below of triangle MN 0, LM and LO are bisected by MS and OR, respectively. Segments MS and OR intersect at T, and m<N = 40°. (image omitted)
If m<TMR = 28°, the measure of angle OTS is

(4) 70°. Again, we can use the Exterior Angle Theorem to figure this out.
TMR = 28, but since MS bisects <M, <TMO is also 28, and <OMN = 56°. Since <N = 40°, <MON = 180 - 40 - 56 = 84°. Since OR bisects <MON, then <MOT and <SOT are each 42°.
Look at triangle MOT. If <TMO = 28 and <MOT = 42, then <OTS = 28 + 42 = 70°.


18.In the diagram below, right triangle ABC has legs whose lengths are 4 and 6. (image omitted)
What is the volume of the three-dimensional object formed by continuously rotating the right triangle around AB?

(1) 32 pi. First, rotating the triangle creates a cone. The Volume of a cone is (1/3)(Area of the Base)(height). The area of the base is (pi)(r)2, where the radius is 4. The height is 6 because AB is the axis the triangle is rotated about.
V = (1/3)(pi)(4)2(6) = (1/3)(16)(pi)(6) = 32 pi.

If you look at the incorrect answers: (2) assumes you used 6 as the radius (rotating about AC), (3) assumes you forgot the 1/3, and (4) assumes that you made both mistakes.


19.What is an equation of a line that is perpendicular to the line whose equation is 2y = 3x - 10 and passes through (-6,1)?

(2) y = -2/3 x - 3. The slope of the line in the question is 3/2. The slope of a perpendicular line must be -2/3 because it is the inverse reciprocal, so choices (3) and (4) are out.
Substitute -6 for x in choices (1) and (2).
(1) y = -2/3 (-6) - 5 = 4 - 5 = -1. No good. We don't want (-6, -1)
(2) y = -2/3 (-6) - 3 = 4 - 3 = 1. Check.


20. In quadrilateral BLUE shown below, BE ~ UL. (image omitted)
Which information would be sufficient to prove quadrilateral BLUE is a parallelogram?

(2) LU || BE. If a pair of line segments in a quadrilateral are both parallel and congruent, then the quadrilateral in a parallelogram. This information could be used in the proof in Part IV.
Note that choices (3) and (4) only show that three sides are congruent, not four. That could be a trapezoid.


21. A ladder 20 feet long leans against a building, forming an angle of 71° with the level ground. To the nearest foot, how high up the wall of the building does the ladder touch the building?

(4) 19 You know the length of the ladder, which is the hypotenuse of the right triangle formed by the ladder, the ground and the wall. The wall is opposite the 71° angle. So you need to use sine to find the answer.

So sin 71° = x / 20
x = 20 sin 71° = 18.91 = 19 feet.


22.In the two distinct acute triangles ABC and DEF, <B = <E. Triangles ABC and DEF are congruent when there is a sequence of rigid motions that maps

As a result of discrepancies in the wording, Questions 22 does not have one clear and correct answer. As a result, all choices were accepted.




23.A fabricator is hired to make a 27-foot-long solid metal railing for the stairs at the local library. The railing is modeled by the diagram below. The railing is 2.5 inches high and 2.5 inches wide and is comprised of a rectangular prism and a half-cylinder. (image omitted)
How much metal, to the nearest cubic inch, will the railing contain?

(1) 151. A reminder: there is no simple one-step formula that will give you this answer!
First you have to differentiate how much of the 2.5 inch height is part of the rectangular prism and what is part of the half-cylinder. The diameter of the half-cylinder is 2.5 in, which means that the radius is 1.25 in. Therefore the height of the prism is 1.25.

So the Volume of the railing = the Volume of the prism + the Volume of the half-cylinder
V = L * W * H + (1/2) (pi) (r)2 h
V = (2.5)(1.25)(27) + (1/2)(pi)(1.25)2(27)
V = 84.375 + 66.268..
V = 150.643... = 151




24.In the diagram below, AC = 7.2 and CE = 2.4. (image omitted)
Which statement is not sufficient to prove triangle ABC ~ triangle EDC?

(2) DE = 2.7 and AB = 8.1. In Choice (1), the parallel lines create alternate interior angles that are congruent, so the triangles are similar by AA (or AAA). The other three choices are show ratios that are 3:1, so the numbers themselves are not important. What is important is which sides are given. Keep in mind that there are vertical angles, so we do have one pair of congruent angles. So the other choices need to give us SAS or SSS. Choice (2) gives us SSA, which isn't allowed.

End of Part I

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Tuesday, July 04, 2017

Happy Fourth of July 2017!

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(C)Copyright 2017, C. Burke.

Can't be more like it than actually being it.

Mike's wife is thinking, "No one better be taking pictures from behind me."
I know this from experience.

I probably should add that I took that photo during a family vacation some years ago, so I'm the copyright holder. It is not in public domain.




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Wednesday, June 28, 2017

Pi vs. Tau

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(C)Copyright 2017, C. Burke.

Would you rather have
Pi in the Sky or Tau in the Scow?




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Monday, June 26, 2017

June 2017: Common Core Algebra Regents, Part 1

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

Answers to Part II can be found here.

Answers to Part III can be found here.

Answers to Part IV can be found here.

June 2017, Algebra I (Common Core), Part I

1.To keep track of his profits, the owner of a carnival booth decided to model his ticket sales on a graph. He found that his profits only declined when he sol between 10 and 40 tickets. Which graph could represent his profits?

(3) (image omitted). If his profits declined, then the y values will decrease in that interval -- in other words, the graph will go down, but only between x = 10 and x = 40. In choice (1), his profits are flat; that is, the remain the same. In choice (4), there is a decline, but it is in the wrong place.


2.The formula for the surface area of a right rectanguar prism is A = 2lw + 2hw + 2lh, where l, w, and h represent the length, width, and height, respectively. Which term of this formula is not dependent on the height?

(3) 2lw. There is no height, h, in the term.


3.Which graph represents y = SQRT(x - 2)?

(4) image omitted. Hopefully, this was an easy one. Besides the fact that you had a graphing calculator available to you, only one of the four graphs was a square root graph. The "- 2" under the radical shifts the graph two units to the right away from the origin.

You should make a note of this. Shifting functions is a common theme and may show up again


4.A student plotted the data from a sleep study as shown in the graph below. (image omitted)
The student used the equation of the line y = -0.09x + 9.24 to model the data. What does the rate of change represent in terms of these data?

(2) The average number of hours of sleep per day increases 0.09 hour per year of age.
Negative correlation. Sleep goes down as age goes up.


5. Lynn, Jude, and Anne were given the function f(x) = -2x2 + 32, and they were asked to find f(3). Lynn's answer was 14, Jude's answer was 4, and Anne's answer was +4. Who is correct?

(1) Lynn, only. Calculate (-2)(3)2 + 32 = (-2)(9) + 32 = -18 + 32 = 14.
The other two tried to solve for the zeroes of the function instead substitution 3 into the function.


6.Which expression is equivalent to 16x4 - 64?

(3) (4x2 + 8)(4x2 - 8). Difference of Squares rule.
If you multiply the choices, choice (1) will have a middle term of -64x2. Choice (2) doesn't even produce the terms in the original expression, plus it will have an x2 term. Choice (4) also yields incorrect coefficients, but no middle term.


7.Vinny collects population data, P(h), about a specific strain of bacteria over time in hours, h, as shown in the graph below. (image omitted)
Which equation represents the graph of P(h)?

(1) P(h) = 4(2)h. If you look at the graph, you'll notice that the y-values are doubling. It's exponential with a base of 2.
If you look at the choices, there is only one exponential function. The others are linear, quadratic and cubic (3rd power).
If you weren't sure if the graph was a parabola, then you could have looked at the y-intercept. In Choice (3), if you substitute 0 for h, then P(h) = 3(0)2 + 0.2(0) + 4.2 = 4.2. However, the graph shows a point at (0, 4), so choice (3) is incorrect.


8.What is the solution to the system of equations below?

y = 2x + 8
3(-2x + y) = 12

(1) no solution. Since "no solutions" and "infinite solutions" are options, it might make sense to rewrite the second equation in slope-intercept form first.

3(-2x + y) = 12
-2x + y = 4
y = 2x + 4

The first equation has a slope of 2 and y-intercept of 8. The second has the same slope but a different y-intercept.
That makes them parallel, and there will be no solutions.


9.A mapping is shown in the diagram below. (image omitted)
This mapping is

(3) not a function, because Feb has two outputs, 28 and 29. Each element in the domain (input) can map to only one element of the range (output). It is okay for Jan and Mar to both map to 31.


10. Which polynomial function has zeroes at -3, 0 and 4?

(3) f(x) = x(x + 3)(x - 4). Flip the signs. Which of the choices will give you a result of 0 when you enter each of those values?
Choices (1) and (2) don't work for 0 (you can stop checking). In choice (4), -3 - 3 =/= 0, nor is 4 + 4.


11.Jordan works for a landscape company during his summer vacation. He is paid $12 per hour for mowing lawns and $14 per hour for planting gardens. He can work a maximum of 40 hours per week, and would like to earn at least $250 this week. if m represents the number of hours mowing lawns and g represents the number of hours planting gardens, which system of inequalities could be used to represent the given conditions?

(1) m + g < 40, 12m + 14g > 250.
The hours must be less then or equal to 40. The amount he makes is the sum of the number of hours times the pay for those hours for each of these jobs, and that should be greater than or equal to 250.


12.Anne invested $1000 in an account with a 1.3% annual interest rate. She made no deposits or withdrawals on the account for 2 years. If interest was compounded annually, which equation represents the balance in the account after the 2 years?

(2) A = 1000(1 + 0.013)2. Interest increases your value, so choices (1) and (3) are right out. The percent 1.3% must be converted to a decimal, which is 0.013. The correct answer is (2).


13. Which value would be a solution for x in the inequality 47 - 4x < 7?

(4) 11. You could solve the inequality, or you can plug in the choices to see what works. Note that 47 - 4(10) = 7, it is not less than 7. Don't let them catch you on that.

47 - 4x < 7
-4x < -40
x > 10
11 is greater than 10.



14.Bella recorded data and used her graphing calculator to find the equation for the line of best fit. She then used the correlation coefficient to determine the strength of the linear fit.
Which correlation coefficient represents the strongest linear relationship?

(1) 0.9. The closer to 1 or -1, the stronger the correlation, the closer to a straight line you get.


15.The height, in inches, of 12 students are listed below
61, 67, 72, 62, 65, 59, 60, 79, 60, 61, 64, 63
Which statement best describes the spread of these data?

(4) 79 is an outlier, which would affect the standard deviation of these data.


16.The graph of a quadratic function is shown below. (image omitted)
An equation that represents the function could be

(4) q(x) = -1/2 (x - 15)2 + 25. The four choices are written in vertex form. The vertex is (15, 25).
Vertex form is y = a(x - h)2 + k. h = 15 and k = 25. (Note that the minus is part of the form.)


17.Which statement is true about the quadratic function g(x), shown in the table below, and f(x) = (x - 3)2 + 2?

(3) They have the same axis of symmetry. The vertex of the equation is (3, 2). The vertex of the table is (3, -5). These points are different but they both lie on the axis of symmetry x = 3.


18.Given the function f(n) defined by the following:

f(1) = 2
f(n) = -5f(n - 1) + 2
Which set could represent the range of this function?

(2) {2, -8, 42, -208, ...}. Frankly, the other choices make no sense. The first number has to be 2, so choices (3) and (4) are right out. Choice (1) is all positive, but the negative multiplier. Substitute 2, 3, and 4, and you'll get the rest of the numbers in the list.


19.An equation is given below.

4(x - 7) = 0.3(x + 2) + 2.11
The solution to the equation is

(1) 8.3

4(x - 7) = 0.3(x + 2) + 2.11
4x - 28 = 0.3x + 0.6 + 2.11
3.7x = 2.71 + 28
3.7x = 30.71
x = 8.3



20.A construction worker needs to move 120 ft3 of dirt by using a wheelbarrow. One wheelbarrow load holds 8 ft3 of dirt and each load takes him 10 minutes to complete. One correct way to figure out the number of hours he would need to complete his job is

(4) (image omitted)
Choice (4) is the only one where all the units cancel out, except for hours.


21.One characteristic of all linear functions is that they change by

(3) equal differences over equal intervals. In other words, a constant rate of change.


22.What are the solutions to the equation x2 - 8x = 10

(2) 4 + SQRT(26)

x2 - 8x = 10
x2 - 8x + 16 = 10 + 16
(x - 4)2 = 26
x - 4 = SQRT(26)
x = 4 + SQRT(26)



23.The formula for blood flow rate is given by F = (P1 - P2) / r, where F is the flow rate, P1 the initial pressure, P2 the final pressure, and r the resistance created by blood vessel size. Which formula can not be derived from the the given formula?

(3) r = F(p2 - p1). To solve for r, you would have to multiply the original equation by r, but then you would have to divide by F, which would put it in the denominator of a fraction.




24.Morgan throws a ball up into the air. The height of the ball above the ground, in feet, is modeled by the function h(t) = -16t2 + 24t, where t represent the time, in seconds, since the ball was thrown. What is the appropriate domain for this situation?

(1) 0 < t < 1.5. The domain is the t axis, so choices (3) and (4) are no good.
If you substitute t = 1.5, then h(1.5) = 0, which is the ground. Everything after that would yield a negative number meaning that the ball went below the ground.

End of Part I

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Sunday, June 25, 2017

June 2017: Common Core Geometry Regents, Part 4

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.

The answers to Part III can be found here.

June 2017, Geometry (Common Core), Part IV

35.Quadrilateral PQRS has vertices P(-2, 3), Q(3, 8), R(4, 1) and S(-1, -4).
Prove that PQRS is a rhombus.
[The use of the set of axes is optional.]

This is NOT a two-column proof. You could write one, but it will not be worth any points if you don’t use the coordinates and the formulas to show the work. You need to back up everything you say with numbers. (And those numbers have to be correct.)

There are a number of ways to show that a quadrilateral is a rhombus. The easiest is that the four sides have equal lengths. You must have a concluding statement that says this and show the work. (Radical 50 or 5 radical 2)

You can show that the diagonals are perpendicular bisectors of each other. So you need to find the slopes of the lines and the midpoint of each line. The slopes must be inverse reciprocals and therefore are perpendicular (that last part is important!) and the midpoints must be the same point.

If you only prove that the slopes are perpendicular, you MUST also show that the figure is a parallelogram. Otherwise, your proof is incomplete. A kite, for example, has perpendicular diagonals.

To show that it is NOT a square, you can show that two sides don’t have right angles: find the slopes and show that they are not perpendicular. You can show that the diagonals are not congruent: find the lengths of the two diagonals. You can even use Pythagorean Theorem to show that two sides and one diagonal do not form a right triangle.

Note that you did not have to simplify your radicals when you use the distance formula, but you did have to have accurate numbers.

Seem the image below.

Points: Basically, you got 4 points for proving and stating the figure was a rhombus and 2 points for proving that it was not a square, for a total of 6 points.




36. Freda, who is training to use a radar system, detects an airplane flying at a constant speed and heading in a straight line to pass directly over her location. She sees the airplane at an angle of elevation of 15o and notes that it is maintaining a constant altitude of 6250 feet. One minute later, she sees the airplane at an angle of elevation of 52o. how far has the airplane traveled, to the nearest foot?

Determine and state the speed of the airplane, to the nearest mile per hour.

You have to find the horizontal distance to the plane from the first sighting, and then the horizontal distance of the second sighting. And then subtract the two values to find how far the plane traveled.
Finally, you have to convert the number of feet it moved per second instead miles per hour.

This could also be done with the Law of Sines but I'll save that for another post, where I can go more in-depth into the problem.

Sketch a figure, or two, to show the plane's location. In each right triangle, you have the height of 6250, which is opposite the angle (either 15 or 52). And you need the ground distance, which will be adjacent to the angle. You don't need the hypotenuse in either triangle. That means using tangent.

So Tan 15o = 6250 / x and Tan 52o = 6250 / y. And the distance traveled will be x - y.


x = 6350 / tan 15 and y = 6250 / tan 52
x = 23325/32 and y = 4883.03
x - y = 18442 feet

No matter what number you got in the top half of the problem, you can get point for converting it into milers per hour correctly.
To change feet per second into miles per hour, multiply by 60 and divide by 5280.

18442.28... x 60 / 5280 = 1106536.94... / 5280 = 297.57... = 210 mph.

You might have run into problems if your calculator was in Radians mode. Hopefully, the negative numbers alerted you to a problem.

If you used sine or cosine (other than using the Law of Sines), you would have lost two points from the top portion of the problem.


End of Part IV

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.