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 . Then , the radius of the sphere is .

Options

A. 6 cm
B. 8 cm
C. 9 cm
D. 3 cm

Answer

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

### Step 1: Calculate the Volume of the 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 of the base and \( h \) is the height of the cone.

Given:
- Height \( h = 24 \) cm
- Radius \( r = 6 \) 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 \( 36 \times 24 \):
\[ 36 \times 24 = 864 \]
Now substituting this value:
\[ V = \frac{1}{3} \pi (864) = 288 \pi \text{ cm}^3 \]

### Step 2: Set the Volume of the Sphere Equal to the Volume of the Cone
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Since the volume of the clay remains the same when reshaped, we set the volume of the cone equal to the volume of the sphere:
\[ \frac{4}{3} \pi r^3 = 288 \pi \]

### Step 3: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = 288 \]
Now, multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 288 \times \frac{3}{4} \]
Calculating \( 288 \times \frac{3}{4} \):
\[ 288 \div 4 = 72 \quad \text{and} \quad 72 \times 3 = 216 \]
Thus, we have:
\[ r^3 = 216 \]

### Step 4: Find the Radius of the Sphere
To find \( r \), we take the cube root of both sides:
\[ r = \sqrt[3]{216} \]
Calculating \( \sqrt[3]{216} \):
\[ 216 = 6^3 \quad \text{(since \( 6 \times 6 \times 6 = 216 \))} \]
Thus:
\[ r = 6 \text{ cm} \]

### Conclusion
The radius of the sphere is \( 6 \) cm.

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