Math: 963da34c

ID: 963da34c 

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)

Comment: Work through one piece of information at a time. Shortcuts can lead to dead ends.

Method 1: The question asks about the allowable width, x, of a box, and since all the lengths are given in inches, we do not need to worry about conversions. Parse the information, sentence by sentence.

Sentence 1: backstory—no help whatsoever

Sentence 2: perimeter of the base + the height ≤ 130

Sentence 3: explains the dimensions included in the perimeter, width and length

Sentence 4: height = 60 inches, length = 2.5 * width, x

Combining 2 and 4,

P (perimeter) + (60) ≤ 130

P ≤ 70

P = 2L + 2W (general equation)

P = 2(2.5x) + 2x

P = 5x + 2x

P = 7x

(7x) ≤ 70

x ≤ 10

This alone should get you to (A) as the correct answer, but take a moment to think about the lower limit. Why do all four answer choices start with 0 < x? It must be less than rather than less than or equal to because the box must have some width, or it would not be a three-dimensional object. The correct answer must be (A).

Method 2: This question is probably best solved algebraically or logically, since there are no ready-made inequalities to plug into Desmos. However, you could create a system by writing one inequality and an equation using the above information. It just takes a little care.

P (perimeter) + (60) ≤ 130

(2L + 2W) ≤ 70

L = 2.5W

You want W to be x, since that is the value the question asks for, so let L be y.

The blue line intersects the red one at a maximum value of 10, so x ≤ 10, and the correct answer is (A).

Method 3: There is not necessarily an easier way to derive the answer, since the bulk of the work you need to do here is to transcribe words into mathematical language. Still, if you were stumped, you could work backwards from the answer choices. Since they all have the same lower limit, you could simply explore whether the supposed maximum allowable limit was, in fact, permissible within the given constraints. It is usually beneficial to start with one of the middle answer choices, since the answers themselves tend to go in increasing or decreasing order. We already know the answer, but test, say, (C) first.

- If the maximum allowable width, x, were 17.5, then the perimeter, knowing that L = 2.5W, would be 2(2.5x) + 2x --> 5x + 2x --> 7x --> 7(17.5) = 122.5. Since 122.5 exceeds the allowable perimeter of 70, you can eliminate (C). Also, (D) will be too great, since 20 > 17.5, meaning there is no need to test.

- Testing (B) in the same manner, you will get 7(11.666) = 81.662, which is also greater than 70. With this option eliminated, the correct answer must be (A). I would not recommend this approach over the others above, but it can help if you blank on how to set up the problem.