A right circular cone is 4.1 cm high and the radius of its base is 2.1 cm. Another right circular cone is 4.3 cm high and the radius of the base is 2.1 cm. Both the cones are melted and recast into a sphere. Find the diameter of the sphere.

Options

A. 6.4 cm
B. 4.2 cm
C. 2.1 cm
D. 5.6 cm

Answer

To find the diameter of the sphere formed by melting the two right circular cones, we will follow these steps:

### Step 1: Calculate the volume of the first cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
For the first cone:
- Radius \( r_1 = 2.1 \, \text{cm} \)
- Height \( h_1 = 4.1 \, \text{cm} \)

Substituting the values:
\[ V_1 = \frac{1}{3} \pi (2.1)^2 (4.1) \]
Calculating \( (2.1)^2 = 4.41 \):
\[ V_1 = \frac{1}{3} \pi (4.41)(4.1) = \frac{1}{3} \pi (18.081) \approx 6.027 \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of the second cone.
For the second cone:
- Radius \( r_2 = 2.1 \, \text{cm} \)
- Height \( h_2 = 4.3 \, \text{cm} \)

Substituting the values:
\[ V_2 = \frac{1}{3} \pi (2.1)^2 (4.3) \]
Using \( (2.1)^2 = 4.41 \):
\[ V_2 = \frac{1}{3} \pi (4.41)(4.3) = \frac{1}{3} \pi (18.963) \approx 6.321 \pi \, \text{cm}^3 \]

### Step 3: Calculate the total volume of the two cones.
Now, we add the volumes of both cones:
\[ V_{\text{total}} = V_1 + V_2 = 6.027 \pi + 6.321 \pi = 12.348 \pi \, \text{cm}^3 \]

### Step 4: Set the total volume equal to the volume of the sphere.
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Setting the total volume equal to the volume of the sphere:
\[ \frac{4}{3} \pi r^3 = 12.348 \pi \]

### Step 5: Simplify and solve for \( r^3 \).
Dividing both sides by \( \pi \):
\[ \frac{4}{3} r^3 = 12.348 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ r^3 = \frac{3 \times 12.348}{4} = 9.261 \]

### Step 6: Find the radius \( r \).
Taking the cube root:
\[ r = \sqrt[3]{9.261} \approx 2.1 \, \text{cm} \]

### Step 7: Calculate the diameter of the sphere.
The diameter \( d \) of the sphere is given by:
\[ d = 2r = 2 \times 2.1 = 4.2 \, \text{cm} \]

Thus, the diameter of the sphere is **4.2 cm**.

---