Find the number of metallic circular discs with 1.5 cm base diameter and of height 0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

Answer

To find the number of metallic circular discs needed to form a right circular cylinder, we will first calculate the volume of both the cylinder and one disc, and then divide the volume of the cylinder by the volume of one disc.

### Step 1: Calculate the Volume of the Cylinder

The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder

Given:
- Diameter of the cylinder = 4.5 cm, so the radius \( r = \frac{4.5}{2} = 2.25 \) cm
- Height of the cylinder \( h = 10 \) cm

Now, substituting the values into the formula:

\[ V = \pi (2.25)^2 (10) \]

Calculating \( (2.25)^2 \):

\[ (2.25)^2 = 5.0625 \]

Now substituting back:

\[ V = \pi (5.0625)(10) = 50.625\pi \, \text{cm}^3 \]

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

The formula for the volume \( V \) of a disc (circular cylinder) is also given by:

\[ V = \pi r_1^2 h_1 \]

Where:
- \( r_1 \) is the radius of the disc
- \( h_1 \) is the height of the disc

Given:
- Diameter of the disc = 1.5 cm, so the radius \( r_1 = \frac{1.5}{2} = 0.75 \) cm
- Height of the disc \( h_1 = 0.2 \) cm

Now substituting the values into the formula:

\[ V = \pi (0.75)^2 (0.2) \]

Calculating \( (0.75)^2 \):

\[ (0.75)^2 = 0.5625 \]

Now substituting back:

\[ V = \pi (0.5625)(0.2) = 0.1125\pi \, \text{cm}^3 \]

### Step 3: Calculate the Number of Discs Needed

To find the number of discs \( n \), we divide the volume of the cylinder by the volume of one disc:

\[ n = \frac{V_{\text{cylinder}}}{V_{\text{disc}}} = \frac{50.625\pi}{0.1125\pi} \]

The \( \pi \) cancels out:

\[ n = \frac{50.625}{0.1125} \]

Calculating this:

\[ n = 450 \]

### Conclusion

The number of metallic circular discs needed to form the right circular cylinder is **450**.

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