SAT Area and volume Practice Question 4 (Hard)

Math · Geometry and TrigonometryHard

Two rectangular prisms are similar. The total surface area of the smaller prism is $54$ square units, and the total surface area of the larger prism is $150$ square units. If the volume of the smaller prism is $27$ cubic units, what is the volume, in cubic units, of the larger prism?

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Answer: 125

The ratio of surface areas is $\frac{54}{150} = \frac{9}{25}$ , which is the square of the ratio of corresponding lengths, so the length ratio is $\frac{3}{5}$ . Volumes scale by the cube of the length ratio: $\frac{27}{125}$ . Since the smaller volume is $27$ , the larger volume is $125$ .

Scaling the volume by the area ratio, $27 \cdot \frac{150}{54} = 75$ , is the most common error.

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