Question

The ratio of volumes of two cubes is \( 1 : 64 \). The ratio of their total surface area is:

• A. \( 1 : 4 \)
• B. \( 1 : 16 \)
• C. \( 1 : 18 \)
• D. \( 1 : 8 \)

Hint:

The ratio of the surface areas of two cubes is the square of the ratio of their side lengths.

Answer 1

The Correct Option is B

Let the side length of the first cube be \( a \) and the side length of the second cube be \( b \). The volume of a cube is given by \( V = a^3 \), and the total surface area of a cube is given by \( A = 6a^2 \). We are given that the ratio of the volumes of the two cubes is \( 1 : 64 \). Thus, we have: \[ \frac{a^3}{b^3} = \frac{1}{64}. \] Taking the cube root of both sides: \[ \frac{a}{b} = \frac{1}{4}. \] Now, the ratio of the total surface areas of the two cubes is: \[ \frac{A_1}{A_2} = \frac{6a^2}{6b^2} = \frac{a^2}{b^2}. \] Since \( \frac{a}{b} = \frac{1}{4} \), we have: \[ \frac{a^2}{b^2} = \left( \frac{1}{4} \right)^2 = \frac{1}{16}. \] Therefore, the ratio of the total surface areas of the two cubes is \( \boxed{1 : 16} \).