Question 166·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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The table gives the volume and surface area of two similar right rectangular prisms, where $k$ is a constant.

	Volume (cubic centimeters)	Surface Area (square centimeters)
Prism $A$	$5{,}832$	$2{,}052$
Prism $B$	$1{,}259{,}712$	$k$

Which choice is the value of $k$ ?

Similar solids have a linear scale factor, the number multiplying each length. Volume uses its cube, while surface area uses its square. Find the volume ratio from A to B, take its cube root to get the length factor, then square that factor for the surface area. Volume and surface area cannot use the same multiplier.

Hints

Hint 1
The volume ratio compares the new prism with the old one. From A to B, put B’s volume over A’s. Volume uses three lengths, so what power of the length factor gives that ratio?
Hint 2
The linear scale factor tells you how much each length grows. Find the number whose cube is the volume ratio. Don’t apply that entire ratio to surface area, which uses two lengths rather than three.
Hint 3
Each face has two lengths, so the surface-area factor is the length factor squared. Multiply prism A’s surface area by that factor to find the surface area of prism B.

Step-by-step

Convert the volume ratio to an area factor
Step 1
Find the volume ratio
Let $s$ be the linear scale factor, the number multiplying each length from prism $A$ to prism $B$ . Volume multiplies three lengths, so the volume ratio equals $s^3$ :
$s^3=\frac{1{,}259{,}712}{5{,}832}.$
Type $r=\frac{1259712}{5832}$ in Desmos. It shows $r=216$ , so $s^3=216$ .
Step 2
Recover the length factor
Find the length factor whose cube is $216$ . Type $x_1^3\sim r$ in Desmos: $x_1$ stands for the unknown factor, and $\sim$ tells Desmos to fit the equality. Under PARAMETERS, Desmos shows $x_1=6$ , so $s=6$ .
Step 3
Set up the surface area
Each face’s area multiplies two lengths, so surface area scales by $s^2$ , not by the volume factor $s^3$ . Starting with prism $A$ ’s $2{,}052$ square centimeters:
$k=2{,}052s^2.$
Step 4
Calculate the requested area
Type $2052x_1^2$ on the next Desmos line. It shows $73{,}872$ . So the surface area of prism $B$ is $73{,}872$ square centimeters. Choice B.

Question 166·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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