SBAC Practice Test Questions 29-30 (Day 175)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The lateral faces of this right prism are congruent squares. If the prism's volume is 10,560.75 cubic inches, what is its height to the nearest inch?

[Here is the given info from the diagram: it's a triangular prism.]

Notice that nowhere in this problem is it directly stated that the bases are equilateral triangles, but we can show that they are indeed equilateral. Here's how -- the question asks for the height of this prism, which we can call h. Then each side of the square lateral faces must be h. But each side of the triangular bases is also a side of a square, hence it must also be h.

So we must find the area of the base -- an equilateral triangle of side h. We see that the equilateral triangle can be divided into two 30-60-90 triangles with hypotenuse h and shorter leg h/2, and so the longer leg -- the altitude of the equilateral triangle -- must be h sqrt(3)/2. Thus the area of the equilateral triangle must be h^2 sqrt(3)/4.

Now we find the volume of the prism:

V = Bh
V = (h)h^2 sqrt(3)/4
V = h^3 sqrt(3)/4 = 10560.75

This is the only way to solve the problem -- we let the desired height be h, find the volume in terms of h, and then solve for h.

h^3 sqrt(3)/4 = 10560.75
h^3 = 10560.75(4)/sqrt(3)
h^3 = 24389.007
h = cbrt(24389.007)
h = 29.000003 inches

Therefore the desired height is 29" -- and of course, today's date is the 29th. Volumes of prisms appear in Lesson 10-5 of the U of Chicago text, but the equilateral triangle area needed to complete the problem can't be completed until special right triangles are studied in Lesson 14-1.

Let's get back to SBAC Prep/Final Exam Prep. Last year I never did cover the last six of the Released Test Questions. I fix this error by finally covering them now.

Question 29 of the SBAC Practice Exam is on angles of elevation:

Emma is standing 10 feet away from the base of a tree and tries to measure the angle of elevation to the top. She is unable to get an accurate measurement, but determines that the angle of elevation is between 55 and 75 degrees.

Decide whether each value given in the table is a reasonable estimate for the tree height. Select Reasonable or Not Reasonable for each height.

                Reasonable  Not Reasonable
4.2 feet
14.7 feet
24.4 feet
33.9 feet
39.1 feet
58.7 feet

Once again, let's use h for the height again. Angle of elevation problems usually depend on the tangent ratio, where h is the height and 10 is the distance to the tree:

tan theta = h/10

But we don't know what the angle of elevation theta is, except that it's between 55 and 75. So let's try solving the problem for both of the extreme values:

tan 55 = h/10
h = 10 tan 55
h = 14.3 feet

tan 75 = h/10
h = 10 tan 75
h = 37.3 feet

And the true height of the tree can be anywhere in between. We thus choose Reasonable for Emma's three heights in this range -- 14.7, 24.4, and 33.9 feet -- and Not Reasonable for her other three -- 4.2, 39.1, and 58.7 feet.

Both the girl and the guy from the Pre-Calc class correctly answer this question. Both of them draw right triangles to help them. The girl writes her work for 55 degrees, but not 75 degrees, while the guy starts to use sine, then corrects himself to tangent. Most likely, both of them enter 10 tan 55 on their calculators, so it was obvious that they needed to enter 10 tan 75 without writing out the work.

Question 30 of the SBAC Practice Exam is on modeling with linear equations:

Emily has a gift certificate for $10 to use at an online store. She can purchase songs for $1 each or episodes of TV shows for $3 each. She wants to spend exactly $10.

Part A
Create an equation to show the relationship between the number of songs, x, Emily can purchase and the number of episodes of TV shows, y, she can purchase.

Part B
Use the Add Point tool to plot all possible combinations of songs and TV shows Emily can purchase.

Since each song is $1 and each episode is $3, it's clear that the equation is 1x + 3y = 10. Notice that I include the coefficient for x even though it is 1, because the SBAC interface for Part A requires a coefficient for both variables.

For Part B, there are four possible solutions -- (1, 3), (4, 2), (7, 1), and (10, 0). These solutions are discrete, but I suspect that the SBAC interface automatically connects the points to form a line -- the graph of the linear equation 1x + 3y = 10.

Both the girl and the guy from the Pre-Calc class correctly answer Part A. But the guy's graph isn't linear, because he miscounts and graphs (6, 1) instead of (7, 1). The girl's graph is linear. But both of them miss the solution (10, 0) -- which is valid, as Emily could have bought 10 songs and no shows.

SBAC Practice Exam Question 29

Common Core Standard:

CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

SBAC Practice Exam Question 30

Common Core Standard:

CCSS.MATH.CONTENT.HSA.CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Commentary: Both the tangent ratio and angles of elevation appear in Lesson 14-3 of the U of Chicago Geometry text. Meanwhile, Lesson 8-8 of the U of Chicago Algebra I text is called "Equations for All Lines." In that lesson, linear equations in standard form Ax + By = C are given, and it's stated that lines in standard form often arise naturally from real situations.