A sphere of radius 5 cm is melted to form a cone with base of same radius. The height (in cm) of the cone is

Options

A. 5
B. 10
C. 20
D. 22

Answer

To solve the problem of finding the height of a cone formed by melting a sphere of radius 5 cm, we can follow these steps:

### Step 1: Write the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Substitute the radius of the sphere into the volume formula.
Given that the radius \( r = 5 \) cm, we can substitute this value into the volume formula:
\[ V = \frac{4}{3} \pi (5)^3 \]
Calculating \( (5)^3 \):
\[ (5)^3 = 125 \]
So, the volume of the sphere becomes:
\[ V = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \text{ cm}^3 \]

### Step 3: Write the formula for the volume of a 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 of the cone and \( h \) is the height of the cone.

### Step 4: Since the cone is formed from the melted sphere, set the volumes equal.
Since the volume of the sphere is equal to the volume of the cone, we can set the two volume formulas equal to each other:
\[ \frac{500}{3} \pi = \frac{1}{3} \pi (5^2) h \]

### Step 5: Substitute the radius of the cone into the volume formula.
The radius of the cone is also \( r = 5 \) cm, so we substitute \( r^2 = (5)^2 = 25 \):
\[ \frac{500}{3} \pi = \frac{1}{3} \pi (25) h \]

### Step 6: Simplify the equation.
We can cancel \( \pi \) from both sides and multiply both sides by 3 to eliminate the fraction:
\[ 500 = 25h \]

### Step 7: Solve for the height \( h \).
Now, divide both sides by 25 to find \( h \):
\[ h = \frac{500}{25} = 20 \text{ cm} \]

### Conclusion
The height of the cone is \( 20 \) cm.

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