Question 34·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Rectangular prism $A$ has edge lengths $x$ , $x+2$ , and $x+4$ , where $x$ is a positive constant. The surface area of prism $A$ is 208 cm $^2$ .

Rectangular prism $B$ is similar to prism $A$ and has surface area 468 cm $^2$ .

Which choice is the volume of prism $B$ , in cubic centimeters?

For similar prisms, two surface areas and a question about volume signal two jobs. Find the first prism’s edge length from its three pairs of faces using a Desmos regression. Then get the length factor from the surface-area ratio and cube it to scale the volume. The tempting trap is applying the surface-area ratio directly to volume.

Hints

Hint 1
A rectangular prism has two faces of each size. Find the three face areas by multiplying pairs of edge lengths, then double their sum. What equation does the given surface area produce?
Hint 2
The surface-area equation can have more than one solution, but an edge length must be positive. In Desmos, use $x_1$ as the unknown and add $\{x_1>0\}$ to keep the solution that fits the prism.
Hint 3
For similar prisms, an edge factor $k$ makes surface area grow by $k^2$ but volume grow by $k^3$ . Find $k$ from the two surface areas, then use it to scale prism A’s volume.

Step-by-step

Find the prism, then scale its volume
Step 1
Count the three pairs of faces
Surface area is the total area of the outside faces. The edge lengths make three face sizes, and there are two faces of each size, so
$2(x(x+2)+x(x+4)+(x+2)(x+4))=208.$
Step 2
Solve for the positive edge length
Type the equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . Desmos uses $\sim$ to find a value rather than graph $x$ ; add $\{x_1>0\}$ because an edge length is positive. Under PARAMETERS, it reports $x_1=4$ , so A’s edges are $4$ , $6$ , and $8$ cm.
Step 3
Find prism A’s volume
A prism’s volume multiplies its three edge lengths. Type $v=x_1(x_1+2)(x_1+4)$ ; Desmos shows $v=192$ . That’s A’s volume in cubic centimeters, not yet B’s.
Step 4
Compare the surface areas
Let $q$ be B’s surface area divided by A’s. Type $q=\frac{468}{208}$ ; Desmos shows $q=2.25=\frac94$ . If $k$ is the factor for each edge from A to B, each face scales in two length directions, so $k^2=q$ .
Step 5
Undo the square to find the edge factor
Type $k=\sqrt q$ ; Desmos shows $k=1.5=\frac32$ . Take the positive square root because edge lengths are positive. Undo the square from surface area before using the length factor for volume.
Step 6
Cube the factor for B’s volume
Volume multiplies three edge lengths, so its factor is $k^3$ , not $q$ . Type $v\cdot k^3$ ; Desmos shows $648$ . Prism B’s volume is $648$ cubic centimeters. Choice C.

Question 34·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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