The number of coins, each 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 determining the number of coins needed 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 (height) of the coin

For our coin:
- Radius \( r_1 = 0.75 \) cm
- Height \( h_1 = 0.2 \) cm

Substituting these values into the formula:
\[ V_1 = \pi (0.75)^2 (0.2) \]
Calculating \( (0.75)^2 \):
\[ (0.75)^2 = 0.5625 \]
Now substituting back:
\[ V_1 = \pi \times 0.5625 \times 0.2 = \pi \times 0.1125 \]

### Step 2: Calculate the volume of the cylinder
The volume \( V \) of the cylinder is calculated using the same formula:
\[ V = \pi r^2 h \]
Where:
- Radius \( r_2 = 3 \) cm
- Height \( h_2 = 8 \) cm

Substituting these values into the formula:
\[ V_2 = \pi (3)^2 (8) \]
Calculating \( (3)^2 \):
\[ (3)^2 = 9 \]
Now substituting back:
\[ V_2 = \pi \times 9 \times 8 = \pi \times 72 \]

### Step 3: Set the volumes equal to find the number of coins
Since the volume of the coins melted will equal the volume of the cylinder, we can set up the equation:
\[ n \cdot V_1 = V_2 \]
Substituting the volumes we calculated:
\[ n \cdot (\pi \times 0.1125) = \pi \times 72 \]

### Step 4: Cancel out \( \pi \) and solve for \( n \)
Dividing both sides by \( \pi \):
\[ n \cdot 0.1125 = 72 \]
Now, solving for \( n \):
\[ n = \frac{72}{0.1125} \]

### Step 5: Calculate \( n \)
To simplify \( \frac{72}{0.1125} \), we can multiply the numerator and denominator by 1000 to eliminate the decimal:
\[ n = \frac{72 \times 1000}{112.5} = \frac{72000}{112.5} \]
Calculating \( 72000 \div 112.5 \):
\[ n = 640 \]

### Conclusion
The number of coins needed to be melted to create the cylinder is \( n = 640 \).