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 metallic cylinder, we will follow these steps:

### Step 1: Calculate the radius of the cylinder
Given the diameter of the cylinder is 16 cm, we can find the radius (r1) using the formula:
\[ r_1 = \frac{\text{Diameter}}{2} = \frac{16 \text{ cm}}{2} = 8 \text{ cm} \]

### Step 2: Calculate the volume of the cylinder
The volume (V_cylinder) of a cylinder is given by the formula:
\[ V_{cylinder} = \pi r_1^2 h_1 \]
Substituting the values we have:
\[ V_{cylinder} = \pi (8 \text{ cm})^2 (9 \text{ cm}) = \pi (64 \text{ cm}^2)(9 \text{ cm}) = 576\pi \text{ cm}^3 \]

### Step 3: Calculate the radius of the sphere
Given the diameter of the sphere is 6 cm, we can find the radius (r2) using the formula:
\[ r_2 = \frac{\text{Diameter}}{2} = \frac{6 \text{ cm}}{2} = 3 \text{ cm} \]

### Step 4: Calculate the volume of the sphere
The volume (V_sphere) of a sphere is given by the formula:
\[ V_{sphere} = \frac{4}{3} \pi r_2^3 \]
Substituting the values we have:
\[ V_{sphere} = \frac{4}{3} \pi (3 \text{ cm})^3 = \frac{4}{3} \pi (27 \text{ cm}^3) = 36\pi \text{ cm}^3 \]

### Step 5: Calculate the number of spheres that can be formed
To find the number of spheres (N), we divide the volume of the cylinder by the volume of one sphere:
\[ N = \frac{V_{cylinder}}{V_{sphere}} = \frac{576\pi \text{ cm}^3}{36\pi \text{ cm}^3} \]
The \(\pi\) cancels out, so we have:
\[ N = \frac{576}{36} = 16 \]

### Conclusion
Thus, the number of spheres that can be formed from the metallic cylinder is **16**.

---