Two right circular cones of equal height of radii of base 3 cm and 4 cm are melted together and made to a solid sphere of radius 5 cm. The height of a cone is

Options

A. 10 cm
B. 20 cm
C. 30 cm
D. 40 cm

Answer

To solve the problem, we need to find the height of the two right circular cones that are melted to form a solid sphere. We will use the formula for the volume of a cone and the volume of a sphere.

### Step-by-Step Solution:

1. **Volume of the Cones**:
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.

For the first cone with radius \( r_1 = 3 \) cm:
\[ V_1 = \frac{1}{3} \pi (3^2) h = \frac{1}{3} \pi (9) h = 3 \pi h \]

For the second cone with radius \( r_2 = 4 \) cm:
\[ V_2 = \frac{1}{3} \pi (4^2) h = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h \]

Total volume of the two cones:
\[ V_{total} = V_1 + V_2 = 3 \pi h + \frac{16}{3} \pi h \]
To combine these, we need a common denominator:
\[ V_{total} = \frac{9}{3} \pi h + \frac{16}{3} \pi h = \frac{25}{3} \pi h \]

2. **Volume of the Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For the sphere with radius \( r = 5 \) cm:
\[ V_{sphere} = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \]

3. **Setting the Volumes Equal**:
Since the volume of the melted cones equals the volume of the sphere:
\[ \frac{25}{3} \pi h = \frac{500}{3} \pi \]
We can cancel \( \pi \) from both sides:
\[ \frac{25}{3} h = \frac{500}{3} \]

4. **Solving for Height \( h \)**:
Multiply both sides by 3 to eliminate the fraction:
\[ 25h = 500 \]
Now, divide both sides by 25:
\[ h = \frac{500}{25} = 20 \text{ cm} \]

### Final Answer:
The height of each cone is \( 20 \) cm.