A metallic cylinder of diameter 16 cm and height 9 cm is melted and recast in to sphere of diameter 6cm .How many such spheres can be formed?

Explanation

To solve the problem of how many spheres can be formed from a melted metallic cylinder, we will follow these steps:

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

- The diameter of the cylinder is 16 cm, so the radius \( r \) is:
\[ r = \frac{16}{2} = 8 \text{ cm} \]
- The height \( h \) of the cylinder is 9 cm.

Now, substituting the values into the volume formula:
\[ V_{\text{cylinder}} = \pi (8)^2 (9) = \pi \times 64 \times 9 = 576\pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of One 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.

- The diameter of the sphere is 6 cm, so the radius \( r \) is:
\[ r = \frac{6}{2} = 3 \text{ cm} \]

Now, substituting the value into the volume formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 36\pi \text{ cm}^3 \]

### Step 3: Calculate the Number of Spheres
To find the number of spheres that can be formed, we divide the volume of the cylinder by the volume of one sphere:
\[ \text{Number of spheres} = \frac{V_{\text{cylinder}}}{V_{\text{sphere}}} = \frac{576\pi}{36\pi} \]

The \( \pi \) cancels out:
\[ \text{Number of spheres} = \frac{576}{36} = 16 \]

### Final Answer
Thus, the number of spheres that can be formed is **16**.

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