A child reshapes a cone made up of clay of height 24 cm and radius 6 cm into a sphere. The radius (in cm) of the sphere is

Options

A. a)6
B. b)12
C. c)24
D. d)28

Answer

To find the radius of the sphere formed from the clay of the cone, we need to 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 cone
- \( h \) is the height of the cone

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

Substituting the values into the formula:

\[ V_{\text{cone}} = \frac{1}{3} \pi (6)^2 (24) \]

Calculating \( (6)^2 \):

\[ (6)^2 = 36 \]

Now substituting back:

\[ V_{\text{cone}} = \frac{1}{3} \pi (36)(24) \]

Calculating \( 36 \times 24 \):

\[ 36 \times 24 = 864 \]

So now we have:

\[ V_{\text{cone}} = \frac{1}{3} \pi (864) \]

Calculating \( \frac{1}{3} \times 864 \):

\[ \frac{864}{3} = 288 \]

Thus, the volume of the cone is:

\[ V_{\text{cone}} = 288\pi \, \text{cm}^3 \]

### Step 2: Set the volume of the sphere equal to the volume of the cone.
The volume of a sphere is given by:

\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

Since the volumes are equal:

\[ 288\pi = \frac{4}{3} \pi r^3 \]

### 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 eliminate the fraction, multiply both sides by \( \frac{3}{4} \):

\[ r^3 = 288 \times \frac{3}{4} \]

Calculating \( 288 \times \frac{3}{4} \):

\[ 288 \times \frac{3}{4} = 216 \]

So we have:

\[ r^3 = 216 \]

### Step 5: Find the radius \( r \).
Taking the cube root of both sides:

\[ r = \sqrt[3]{216} \]

Calculating \( \sqrt[3]{216} \):

\[ r = 6 \, \text{cm} \]

### Final Answer:
The radius of the sphere is \( 6 \) cm.

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