Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is :
Show Hint
For any similar 3D shapes, the ratio of areas is always the cube root of the volume ratio squared:
\[ \text{Area Ratio} = (\sqrt[3]{\text{Volume Ratio}})^2 \]
With a volume ratio of \(64:27\), take the cube root to get \(4:3\), then square it to get \(16:9\) instantly.
Step 1: Spot the cubes hiding inside the ratio. $64 = 4^3$ and $27 = 3^3$, which strongly suggests the two actual radii are in the ratio $4:3$. Let us just take the radii to be $4$ units and $3$ units and check everything directly. Step 2: Verify the volume ratio with these radii. \[ V_1 = \frac{4}{3}\pi(4)^3 = \frac{256}{3}\pi, \qquad V_2 = \frac{4}{3}\pi(3)^3 = 36\pi \] \[ \frac{V_1}{V_2} = \frac{256\pi/3}{36\pi} = \frac{256}{108} = \frac{64}{27} \ \checkmark \] Step 3: Now compute the surface areas with the same radii. \[ A_1 = 4\pi(4)^2 = 64\pi, \qquad A_2 = 4\pi(3)^2 = 36\pi \] Step 4: Take the ratio and simplify. \[ \frac{A_1}{A_2} = \frac{64\pi}{36\pi} = \frac{16}{9} \] \[ \boxed{16 : 9} \]