A certain number of metallic cones, each of radius 2 cm and height 3 cm, are melted and recast into a solid sphere of radius 6 cm. Find the number of cones used.

Answer

To solve the problem of finding the number of metallic cones melted and recast into a solid sphere, we will follow these steps:

### Step 1: Understand the problem
We need to find the number of cones (let's denote it as \( n \)) that, when melted, will have the same volume as a solid sphere of radius 6 cm.

### Step 2: Write the formula for the volume of a cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where \( r \) is the radius and \( h \) is the height of the cone.

### Step 3: Calculate the volume of one cone
Given that the radius \( r = 2 \) cm and the height \( h = 3 \) cm, we can substitute these values into the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (3) \]
Calculating this gives:
\[ V_{\text{cone}} = \frac{1}{3} \pi (4)(3) = \frac{12}{3} \pi = 4\pi \text{ cm}^3 \]

### Step 4: Write the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the sphere.

### Step 5: Calculate the volume of the sphere
Given that the radius \( r = 6 \) cm, we substitute this value into the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 \]
Calculating this gives:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288\pi \text{ cm}^3 \]

### Step 6: Set up the equation
Since the total volume of \( n \) cones is equal to the volume of the sphere, we can set up the equation:
\[ n \cdot V_{\text{cone}} = V_{\text{sphere}} \]
Substituting the volumes we calculated:
\[ n \cdot 4\pi = 288\pi \]

### Step 7: Solve for \( n \)
To find \( n \), we can divide both sides of the equation by \( 4\pi \):
\[ n = \frac{288\pi}{4\pi} \]
The \( \pi \) cancels out:
\[ n = \frac{288}{4} = 72 \]

### Conclusion
The number of cones used is \( n = 72 \).

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