How many coins of 1.4 cm in diameter and 0.4 cm thick are to be melted to form a right circular cylinder of height 16 cm and diameter 3.5 cm?

Options

A. 400
B. 250
C. 180
D. 230

Answer

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

### Step 1: Calculate the volume of one coin.
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.

1. **Find the radius of the coin**: The diameter of the coin is 1.4 cm, so the radius \( r \) is:
\[ r = \frac{1.4}{2} = 0.7 \text{ cm} \]

2. **Height of the coin**: The thickness of the coin is given as 0.4 cm, so \( h = 0.4 \text{ cm} \).

3. **Calculate the volume of the coin**:
\[ V_{\text{coin}} = \pi (0.7)^2 (0.4) = \pi \times 0.49 \times 0.4 = 0.196\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the right circular cylinder.
1. **Find the radius of the cylinder**: The diameter of the cylinder is 3.5 cm, so the radius \( R \) is:
\[ R = \frac{3.5}{2} = 1.75 \text{ cm} \]

2. **Height of the cylinder**: The height \( H \) is given as 16 cm.

3. **Calculate the volume of the cylinder**:
\[ V_{\text{cylinder}} = \pi (1.75)^2 (16) = \pi \times 3.0625 \times 16 = 49 \pi \text{ cm}^3 \]

### Step 3: Find the number of coins needed.
To find the number of coins needed, we divide the volume of the cylinder by the volume of one coin:
\[ \text{Number of coins} = \frac{V_{\text{cylinder}}}{V_{\text{coin}}} = \frac{49\pi}{0.196\pi} \]
The \( \pi \) cancels out:
\[ \text{Number of coins} = \frac{49}{0.196} \]

### Step 4: Calculate the final result.
1. **Perform the division**:
\[ \frac{49}{0.196} = 250 \]

Thus, the number of coins needed is **250**.