A cone of height 24 cm and radius of base 6 cm is made up of modeling clay, A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the 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 and \( h \) is the height.

Given:
- Height \( h = 24 \) cm
- Radius \( r = 6 \) cm

Substituting the values into the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (6)^2 (24) \]
Calculating \( (6)^2 = 36 \):
\[ V_{\text{cone}} = \frac{1}{3} \pi (36)(24) \]
Calculating \( 36 \times 24 = 864 \):
\[ V_{\text{cone}} = \frac{1}{3} \pi (864) \]
Calculating \( \frac{864}{3} = 288 \):
\[ V_{\text{cone}} = 288 \pi \text{ cm}^3 \]

### Step 2: Set the Volume of the Sphere Equal to the Volume of the Cone
The volume of the sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Since the cone is reshaped into a sphere, we have:
\[ V_{\text{cone}} = V_{\text{sphere}} \]
Thus,
\[ 288 \pi = \frac{4}{3} \pi r^3 \]

### Step 3: Cancel \(\pi\) from Both Sides
Dividing both sides by \(\pi\):
\[ 288 = \frac{4}{3} r^3 \]

### Step 4: Solve for \( r^3 \)
To isolate \( r^3 \), multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 288 \times \frac{3}{4} \]
Calculating \( 288 \times \frac{3}{4} = 216 \):
\[ r^3 = 216 \]

### Step 5: Find the Radius of the Sphere
Taking the cube root of both sides:
\[ r = \sqrt[3]{216} \]
Calculating the cube root:
\[ r = 6 \text{ cm} \]

### Step 6: Calculate the Surface Area of the Sphere
The formula for the surface area of a sphere is:
\[ A = 4 \pi r^2 \]
Substituting the radius \( r = 6 \) cm:
\[ A = 4 \pi (6)^2 \]
Calculating \( (6)^2 = 36 \):
\[ A = 4 \pi (36) \]
Calculating \( 4 \times 36 = 144 \):
\[ A = 144 \pi \text{ cm}^2 \]

### Final Answer
The radius of the sphere is \( 6 \) cm and the surface area of the sphere is \( 144 \pi \text{ cm}^2 \).

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