A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.

Answer

To find the radius of the sphere formed from the cone made of modeling clay, we need to follow these steps:

### Step 1: Calculate the volume of the cone.
The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

where \( r \) is the radius of the base and \( h \) is the height.

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

Substituting the values into the formula:

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

Calculating \( (6)^2 \):

\[ (6)^2 = 36 \]

Now substituting back:

\[ V = \frac{1}{3} \pi (36)(24) \]

Calculating \( 36 \times 24 \):

\[ 36 \times 24 = 864 \]

So, the volume of the cone becomes:

\[ V = \frac{1}{3} \pi (864) = 288 \pi \text{ cm}^3 \]

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

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

Since the volume of the clay remains the same when reshaped into a sphere, we set the volume of the sphere equal to the volume of the cone:

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

### Step 3: Simplify the equation.
We can cancel \( \pi \) from both sides:

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

### Step 4: Solve for \( r^3 \).
Multiplying 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 \( r \) by taking the cube root.
Now, we take the cube root of both sides:

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

Calculating the cube root:

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

### Conclusion:
The radius of the sphere formed from the cone is \( 6 \) cm.

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