Question 49·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Two closed right circular cylinders, Cylinder A and Cylinder B, are similar.

The total surface area of Cylinder A is $500\pi$ square centimeters, and the total surface area of Cylinder B is $1,280\pi$ square centimeters. The volume of Cylinder A is $250\pi$ cubic centimeters.

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

Similar solids have a linear scale factor, the number multiplying every matching length. A surface area ratio gives the square of that factor, while a volume ratio uses its cube. Divide B’s surface area by A’s, then use Desmos to find the positive length factor. Using the area ratio directly on the volume misses a dimension.

Hints

Hint 1
Similar cylinders have a linear scale factor $k$ : every length in B is $k$ times its match in A. Surface area scales by $k^2$ . Which area should go on top of the ratio when moving from A to B?
Hint 2
An area ratio compares the two surface areas. Divide B’s area by A’s, cancel $\pi$ , and find the positive square root. A length factor can’t be negative.
Hint 3
Volume fills three dimensions, so it scales by the cube of the length factor, not its square. Once you know $k$ , what should multiply Cylinder A’s volume?

Step-by-step

Find the length factor, then scale the volume
Step 1
Turn the surface areas into a scale-factor equation
Let $k$ be the linear scale factor from A to B: every matching length in B is $k$ times as long. Because the cylinders are similar, surface area scales by $k^2$ . Put B over A to keep the direction right:
$k^2=\frac{\text{surface area of B}}{\text{surface area of A}}=\frac{1{,}280\pi}{500\pi}.$
Step 2
Find the positive length factor in Desmos
Cancel $\pi$ from the ratio:
$k^2=\frac{1{,}280}{500}.$
Type $x_1^2\sim1280/500\{x_1>0\}$ , using $x_1$ for $k$ . The $\sim$ asks Desmos to find a value that fits the equation, and the restriction keeps the positive value because lengths can’t be negative. Under PARAMETERS, Desmos shows $x_1=1.6$ , so $k=1.6$ .
Step 3
Choose the factor for volume
Volume fills three dimensions, so scaling each length by $k$ scales the volume by $k^3$ . Surface area scales by $k^2$ ; volume scales by $k^3$ . Use Cylinder A’s given volume:
$V_B=(250\pi)k^3.$
Multiplying by $k$ alone would scale only one dimension.
Step 4
Calculate Cylinder B’s volume
Keep the earlier Desmos line and type $250x_1^3$ to find the number multiplying $\pi$ . Desmos prints $1024$ . Keep the $\pi$ from Cylinder A’s volume, so Cylinder B’s volume is $1024\pi$ cubic centimeters. Choice C.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse scale factors in two and three dimensions

Question 49·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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