Find the number of coins , `3` cm in diameter and `1` cm thickness to be melted to form a right -circular cylinder of height `10` cm and diameter `9` cm .

Options

A. `90`
B. `60`
C. `75`
D. `30`

Answer

To solve the problem of finding the number of coins that need to be melted to form a right circular cylinder, we will follow these steps:

### Step 1: Find the volume of the right circular cylinder formed.
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

- Given:
- Height \( h = 10 \) cm
- Diameter \( d = 9 \) cm, so the radius \( r = \frac{d}{2} = \frac{9}{2} = 4.5 \) cm

Now, substituting the values into the volume formula:
\[ V = \pi \left(4.5\right)^2 \times 10 \]
Calculating \( (4.5)^2 \):
\[ (4.5)^2 = 20.25 \]
Now substituting back:
\[ V = \pi \times 20.25 \times 10 = 202.5\pi \, \text{cm}^3 \]

### Step 2: Find the volume of one coin.
Using the same volume formula for the coin:
- Given:
- Thickness (height) \( h = 1 \) cm
- Diameter \( d = 3 \) cm, so the radius \( r = \frac{d}{2} = \frac{3}{2} = 1.5 \) cm

Now substituting the values into the volume formula:
\[ V_{\text{coin}} = \pi \left(1.5\right)^2 \times 1 \]
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]
Now substituting back:
\[ V_{\text{coin}} = \pi \times 2.25 \times 1 = 2.25\pi \, \text{cm}^3 \]

### Step 3: Find the number of coins that can be formed.
Let \( x \) be the number of coins. The total volume of the coins will equal the volume of the cylinder:
\[ x \times V_{\text{coin}} = V \]
Substituting the volumes we found:
\[ x \times 2.25\pi = 202.5\pi \]
Dividing both sides by \( \pi \):
\[ x \times 2.25 = 202.5 \]
Now, solving for \( x \):
\[ x = \frac{202.5}{2.25} \]
Calculating \( \frac{202.5}{2.25} \):
\[ x = 90 \]

### Final Answer:
The number of coins that need to be melted is \( 90 \).

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