The number of coins of radius `0.75` cm and thickness `0.2` cm to be melted to make a right circular cylinder of height 8 cm and base radius 3 cm is

Options

A. 460
B. 500
C. 600
D. 640

Answer

To solve the problem of how many coins need to be melted to create a right circular cylinder, we will follow these steps:

### Step 1: Calculate the volume of one coin.
The volume \( V \) of a cylinder (which is the shape of the coin) is given by the formula:

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

Where:
- \( r \) is the radius of the coin,
- \( h \) is the thickness of the coin.

Given:
- Radius of the coin \( r = 0.75 \) cm,
- Thickness of the coin \( h = 0.2 \) cm.

Substituting the values into the formula:

\[ V_{coin} = \pi (0.75)^2 (0.2) \]

Calculating \( (0.75)^2 \):

\[ (0.75)^2 = 0.5625 \]

Now substituting back:

\[ V_{coin} = \pi (0.5625)(0.2) = \pi (0.1125) \]

Using \( \pi \approx 3.14 \):

\[ V_{coin} \approx 3.14 \times 0.1125 \approx 0.35325 \text{ cm}^3 \]

### Step 2: Calculate the volume of the right circular cylinder.
The volume \( V \) of a cylinder is also given by:

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

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

Given:
- Radius of the cylinder \( r = 3 \) cm,
- Height of the cylinder \( h = 8 \) cm.

Substituting the values into the formula:

\[ V_{cylinder} = \pi (3)^2 (8) \]

Calculating \( (3)^2 \):

\[ (3)^2 = 9 \]

Now substituting back:

\[ V_{cylinder} = \pi (9)(8) = \pi (72) \]

Using \( \pi \approx 3.14 \):

\[ V_{cylinder} \approx 3.14 \times 72 \approx 226.08 \text{ cm}^3 \]

### Step 3: Calculate the number of coins needed.
To find the number of coins \( n \) required to make the cylinder, we divide the volume of the cylinder by the volume of one coin:

\[ n = \frac{V_{cylinder}}{V_{coin}} = \frac{226.08}{0.35325} \]

Calculating this:

\[ n \approx 640.02 \]

Since we cannot have a fraction of a coin, we round up to the nearest whole number:

\[ n = 641 \]

### Final Answer:
The number of coins needed to be melted to make the right circular cylinder is **641**.

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