A solid right circular metal cylinder with radius 12m and height 15m is melted down and all the metal is used to recast a new solid cylinder with radius 10m. What is the curved surface area of the new cylinder? (in `m^2`)

Options

A. 416
B. 454
C. 448
D. None of these

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the volume of the original cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
For the original cylinder:
- Radius \( r = 12 \) m
- Height \( h = 15 \) m

Substituting the values:
\[ V = \pi (12^2) (15) = \pi (144) (15) = 2160\pi \, \text{m}^3 \]

### Step 2: Set the volume of the new cylinder equal to the volume of the original cylinder.
The volume of the new cylinder is also given by the same formula:
\[ V = \pi r^2 h \]
For the new cylinder:
- Radius \( r = 10 \) m
- Height \( h \) (unknown)

Setting the volumes equal:
\[ 2160\pi = \pi (10^2) h \]
This simplifies to:
\[ 2160 = 100h \]

### Step 3: Solve for the height \( h \) of the new cylinder.
Rearranging the equation:
\[ h = \frac{2160}{100} = 21.6 \, \text{m} \]

### Step 4: Calculate the curved surface area of the new cylinder.
The curved surface area \( A \) of a cylinder is given by the formula:
\[ A = 2\pi rh \]
Substituting the values for the new cylinder:
- Radius \( r = 10 \) m
- Height \( h = 21.6 \) m

Calculating the curved surface area:
\[ A = 2\pi (10)(21.6) = 216\pi \, \text{m}^2 \]

### Step 5: Approximate the value of the curved surface area.
Using \( \pi \approx 3.14 \):
\[ A \approx 216 \times 3.14 \approx 678.24 \, \text{m}^2 \]

### Final Answer:
The curved surface area of the new cylinder is approximately \( 678.24 \, \text{m}^2 \).

---