The number of coins 1.5 cm in diameter and 0.2 cm thick to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm is

Options

A. 540
B. 450
C. 380
D. 472

Answer

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

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
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 \( h = 10 \) cm.

Substituting the values into the formula:
\[ V = \pi (2.25)^2 (10) \]
Calculating \( (2.25)^2 \):
\[ (2.25)^2 = 5.0625 \]
Now substituting back:
\[ V = \pi \times 5.0625 \times 10 = 50.625\pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One Coin
The volume \( V_{coin} \) of a coin (which is also a cylinder) is given by the same formula:
\[ V_{coin} = \pi r_{coin}^2 h_{coin} \]
where:
- Diameter of the coin = 1.5 cm, so the radius \( r_{coin} = \frac{1.5}{2} = 0.75 \) cm.
- Thickness \( h_{coin} = 0.2 \) 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:
\[ V_{coin} = \pi \times 0.5625 \times 0.2 = 0.1125\pi \, \text{cm}^3 \]

### Step 3: Calculate the Number of Coins
Let \( n \) be the number of coins needed. The total volume of \( n \) coins must equal the volume of the cylinder:
\[ n \times V_{coin} = V \]
Substituting the volumes:
\[ n \times 0.1125\pi = 50.625\pi \]
Dividing both sides by \( \pi \):
\[ n \times 0.1125 = 50.625 \]
Now, solving for \( n \):
\[ n = \frac{50.625}{0.1125} \]
Calculating \( n \):
\[ n = 450 \]

### Final Answer
The number of coins that need to be melted to form the cylinder is **450**.

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