A right circular cone is of height 8.4 cm and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is:

Question

Accepted Answer

To determine the radius of the sphere formed by melting and recasting the right circular cone, we need to equate the volume of the cone to the volume of the sphere.

1. **Volume of the Cone:**
   The formula for the volume of a cone is given by:
   $$
   V_{\text{cone}} = \frac{1}{3} \pi r^2 h
   $$
   where $ r $ is the radius of the base of the cone, and $ h $ is the height of the cone.

For the given cone:
   - Radius $ r = 2.1 $ cm
   - Height $ h = 8.4 $ cm

Substituting these values into the formula:
   $$
   V_{\text{cone}} = \frac{1}{3} \pi (2.1)^2 (8.4)
   $$
   $$
   V_{\text{cone}} = \frac{1}{3} \pi \times 4.41 \times 8.4
   $$
   $$
   V_{\text{cone}} = \frac{1}{3} \pi \times 37.044
   $$
   $$
   V_{\text{cone}} = 12.348 \pi \, \text{cm}^3
   $$

2. **Volume of the Sphere:**
   The formula for the volume of a sphere is:
   $$
   V_{\text{sphere}} = \frac{4}{3} \pi R^3
   $$
   where $ R $ is the radius of the sphere.

Since the volume of the cone is equal to the volume of the sphere:
   $$
   \frac{4}{3} \pi R^3 = 12.348 \pi
   $$

Simplifying, we get:
   $$
   \frac{4}{3} R^3 = 12.348
   $$
   $$
   R^3 = \frac{12.348 \times 3}{4}
   $$
   $$
   R^3 = 9.261
   $$

Taking the cube root of both sides:
   $$
   R = \sqrt[3]{9.261} \approx 2.1 \, \text{cm}
   $$

Thus, the radius of the sphere is approximately 2.1 cm, which corresponds to Option 3.

**Explanation for Other Options:**

- **Option 1 (2.3 cm):** If the radius were 2.3 cm, the volume of the sphere would be greater than the volume of the cone, which contradicts the condition that the cone is melted and recast into the sphere.
  
- **Option 2 (2.5 cm):** Similarly, a radius of 2.5 cm would result in an even larger volume for the sphere, which is not possible given the initial volume of the cone.

- **Option 4 (2.7 cm):** A radius of 2.7 cm would yield a significantly larger volume for the sphere, making it impossible to be formed from the given cone.

Therefore, the correct answer is indeed Option 3, with a radius of 2.1 cm for the sphere.