If n coins each of diameter 1.5 cm and thickness 0.2 cm are melted and a right circular cylinder of height 10 cm and diameter 5 cm is made, then n is equal to

Options

A. 336
B. 450
C. 512
D. 555

Answer

To solve the problem, we need to find the number of coins \( n \) that can be melted to form a right circular cylinder. We will use the formula for the volume of a cylinder and equate the total volume of the coins to the volume of the cylinder.

### Step 1: Calculate the volume of one coin
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or thickness in the case of a coin).

For one coin:
- Diameter = 1.5 cm, so the radius \( r \) is:
\[ r = \frac{1.5}{2} = 0.75 \text{ cm} \]
- Thickness \( h \) = 0.2 cm.

Now, substituting these values into the volume formula:
\[ V_{\text{coin}} = \pi (0.75)^2 (0.2) \]
Calculating \( (0.75)^2 \):
\[ (0.75)^2 = 0.5625 \]
Thus, the volume of one coin is:
\[ V_{\text{coin}} = \pi \times 0.5625 \times 0.2 = \pi \times 0.1125 \]

### Step 2: Calculate the volume of \( n \) coins
The total volume of \( n \) coins is:
\[ V_{\text{total}} = n \times V_{\text{coin}} = n \times \pi \times 0.1125 \]

### Step 3: Calculate the volume of the cylinder
Now, we calculate the volume of the cylinder:
- Diameter = 5 cm, so the radius \( r \) is:
\[ r = \frac{5}{2} = 2.5 \text{ cm} \]
- Height \( h \) = 10 cm.

Using the volume formula for the cylinder:
\[ V_{\text{cylinder}} = \pi (2.5)^2 (10) \]
Calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Thus, the volume of the cylinder is:
\[ V_{\text{cylinder}} = \pi \times 6.25 \times 10 = 62.5\pi \]

### Step 4: Set the volumes equal to each other
Since the total volume of the coins equals the volume of the cylinder, we have:
\[ n \times \pi \times 0.1125 = 62.5\pi \]
Dividing both sides by \( \pi \):
\[ n \times 0.1125 = 62.5 \]

### Step 5: Solve for \( n \)
Now, we can solve for \( n \):
\[ n = \frac{62.5}{0.1125} \]
Calculating the right-hand side:
\[ n = \frac{625}{1.125} = 555.555\ldots \]
Thus, rounding to the nearest whole number, we find:
\[ n \approx 556 \]

### Final Answer
Therefore, the value of \( n \) is approximately \( 556 \).