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

Options

A. 6
B. 12
C. 24
D. 48

Answer

To find the radius of the sphere formed by reshaping a cone, we need to equate the volumes of the cone and the sphere. Here’s the step-by-step solution:

### 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 and \( h \) is the height of the cone.

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

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (6)^2 (24) \]

### Step 2: Simplify the volume calculation.
Calculating \( (6)^2 \):
\[ (6)^2 = 36 \]
Now substituting back:
\[ V = \frac{1}{3} \pi (36) (24) \]

### Step 3: Multiply the values.
Calculating \( 36 \times 24 \):
\[ 36 \times 24 = 864 \]
Now substituting back:
\[ V = \frac{1}{3} \pi (864) \]

### Step 4: Divide by 3.
Calculating \( \frac{864}{3} \):
\[ \frac{864}{3} = 288 \]
Thus, the volume of the cone is:
\[ V = 288 \pi \text{ cm}^3 \]

### Step 5: Set the volume of the cone equal to the volume of the sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Setting the volumes equal:
\[ 288 \pi = \frac{4}{3} \pi r^3 \]

### Step 6: Cancel \( \pi \) from both sides.
\[ 288 = \frac{4}{3} r^3 \]

### Step 7: Multiply both sides by \( \frac{3}{4} \).
\[ 288 \times \frac{3}{4} = r^3 \]
Calculating \( 288 \times \frac{3}{4} \):
\[ 288 \times \frac{3}{4} = 216 \]
Thus:
\[ r^3 = 216 \]

### Step 8: Take the cube root.
To find \( r \):
\[ r = \sqrt[3]{216} \]

### Step 9: Calculate the cube root.
The cube root of 216 is:
\[ r = 6 \text{ cm} \]

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

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