The volumes of two cubes are in the ratio 27 : 64. What is the ratio of their…

Concept

For similar solids, volume varies as the cube of a corresponding length, while total surface area varies as the square of that length.

Therefore, first obtain the linear scale from the volume ratio, then use its square for the surface-area ratio.

Application

1. Let the side lengths of the two cubes be a and b. Since their volume ratio is 27 : 64, a3 : b3 = 27 : 64.

2. Taking cube roots of both terms gives a : b = cube root of 27 : cube root of 64 = 3 : 4.

3. Total surface area is 6 times side squared. Therefore, the area ratio is 6a2 : 6b2 = a2 : b2 = 32 : 42 = 9 : 16.

Cross-check

Choose side lengths 3 units and 4 units. Their volumes are 27 and 64 cubic units, and their total surface areas are 54 and 96 square units; 54 : 96 simplifies to 9 : 16.

Result

The ratio of the total surface areas is 9 : 16.