A solid spherical metal ball of diameter 72 cm is melted and recast into small solid cones of diameter 6 cm and height 6 cm. Find the number of cones that can be formed using this melted metal.
A)3456 cones
B)3600 cones
C)3568 cones
D)3200 cones

Options

A. 3456 cones
B. 3600 cones
C. 3568 cones
D. 3200 cones

Answer

To solve the problem of finding the number of small cones that can be formed from a melted solid spherical metal ball, we will follow these steps:

### Step 1: Calculate the volume of the spherical metal ball.
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 the diameter of the sphere is 72 cm, the radius \( r \) is:
\[ r = \frac{72}{2} = 36 \text{ cm} \]

Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (36)^3 \]

Calculating \( 36^3 \):
\[ 36^3 = 36 \times 36 \times 36 = 46656 \]
Thus, the volume of the sphere becomes:
\[ V = \frac{4}{3} \pi (46656) = \frac{186624}{3} \pi = 62208 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one small 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 the diameter of the cone is 6 cm, the radius \( r \) is:
\[ r = \frac{6}{2} = 3 \text{ cm} \]
The height \( h \) of the cone is 6 cm.

Now substituting these values into the volume formula:
\[ V = \frac{1}{3} \pi (3)^2 (6) \]
Calculating \( (3)^2 \):
\[ (3)^2 = 9 \]
Thus, the volume of the cone becomes:
\[ V = \frac{1}{3} \pi (9)(6) = \frac{54}{3} \pi = 18 \pi \text{ cm}^3 \]

### Step 3: Find the number of cones that can be formed.
Let \( n \) be the number of cones that can be formed. Since the volume of the melted sphere is equal to the total volume of the cones formed, we have:
\[ 62208 \pi = n \times 18 \pi \]

Dividing both sides by \( \pi \):
\[ 62208 = n \times 18 \]

Now, solving for \( n \):
\[ n = \frac{62208}{18} \]

Calculating \( \frac{62208}{18} \):
\[ n = 3456 \]

### Conclusion
The number of cones that can be formed using the melted metal is \( \boxed{3456} \).