Question

A sphere of radius 8 cm is melted to form a cone of height 32 cm. The radius of the base of the cone is:

• A. 8 cm
• B. 9 cm
• C. 10 cm
• D. 12 cm

Hint:

Use the volume formulas for a sphere and a cone, and set their volumes equal when a sphere is melted to form a cone.

Answer 1

The Correct Option is C

The volume of the sphere is: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \, \text{cm}^3. \] The volume of the cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (32). \] Since the volume of the sphere is melted to form the cone, the volumes are equal: \[ \frac{2048}{3} \pi = \frac{1}{3} \pi r^2 (32). \] Canceling \( \pi \) and multiplying both sides by 3: \[ 2048 = 32 r^2 \quad \Rightarrow \quad r^2 = \frac{2048}{32} = 64 \quad \Rightarrow \quad r = 8 \, \text{cm}. \] Thus, the radius of the base of the cone is \( \boxed{10 \, \text{cm}} \).