If a metallic cone of radius 30 cm and height 45 cm is melted and recast into metallic spheres of radius 5 cm, find the number of spheres.

Options

A. 81
B. 41
C. 80
D. 40

Answer

To solve the problem of finding the number of metallic spheres that can be formed from a melted metallic cone, we will 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 and \( h \) is the height of the cone.

Given:
- Radius \( r = 30 \) cm
- Height \( h = 45 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (30)^2 (45) \]
Calculating \( (30)^2 = 900 \):
\[ V = \frac{1}{3} \pi (900)(45) \]
Calculating \( 900 \times 45 = 40500 \):
\[ V = \frac{1}{3} \pi (40500) \]
Now, dividing \( 40500 \) by \( 3 \):
\[ V = 13500 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given:
- Radius \( r = 5 \) cm

Substituting the value into the formula:
\[ V = \frac{4}{3} \pi (5)^3 \]
Calculating \( (5)^3 = 125 \):
\[ V = \frac{4}{3} \pi (125) \]
Calculating \( \frac{4 \times 125}{3} = \frac{500}{3} \):
\[ V = \frac{500}{3} \pi \text{ cm}^3 \]

### Step 3: Find the number of spheres.
Let the number of spheres be \( x \). The total volume of the spheres must equal the volume of the cone:
\[ 13500 \pi = x \left(\frac{500}{3} \pi\right) \]
Dividing both sides by \( \pi \):
\[ 13500 = x \left(\frac{500}{3}\right) \]
Multiplying both sides by \( 3 \):
\[ 40500 = 500x \]
Now, dividing both sides by \( 500 \):
\[ x = \frac{40500}{500} = 81 \]

### Conclusion
The number of metallic spheres that can be formed is \( \boxed{81} \).

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