The cost of painting the total outside surface of a closed cylinder at Rs 3 per `cm^(2)` is Rs 2772. If the height of the cylinder is 2 times the radius , then find its volume .

Options

A. `34,312 cm^(3)`
B. `3342 cm^(3)`
C. `2154 cm^(3)`
D. `2156 cm^(3)`

Answer

To solve the problem, we need to find the volume of a closed cylinder given the cost of painting its surface area and the relationship between its height and radius. Let's break it down step by step.

### Step 1: Understand the given information
- The cost of painting the total outside surface of a closed cylinder is Rs 3 per cm².
- The total cost for painting is Rs 2772.
- The height (h) of the cylinder is 2 times the radius (r): \( h = 2r \).

### Step 2: Calculate the total surface area of the cylinder
The total surface area (A) of a closed cylinder is given by the formula:
\[ A = 2\pi r^2 + 2\pi rh \]
Substituting \( h = 2r \):
\[ A = 2\pi r^2 + 2\pi r(2r) = 2\pi r^2 + 4\pi r^2 = 6\pi r^2 \]

### Step 3: Relate the surface area to the cost
The cost of painting the surface area is given by:
\[ \text{Cost} = \text{Area} \times \text{Cost per cm}^2 \]
Substituting the values we have:
\[ 2772 = 6\pi r^2 \times 3 \]
This simplifies to:
\[ 2772 = 18\pi r^2 \]

### Step 4: Solve for \( \pi r^2 \)
To isolate \( \pi r^2 \), divide both sides by 18:
\[ \pi r^2 = \frac{2772}{18} \]
Calculating the right side:
\[ \pi r^2 = 154 \]

### Step 5: Find the radius
Now, we can find \( r^2 \):
\[ r^2 = \frac{154}{\pi} \]

### Step 6: Calculate the height
Since \( h = 2r \), we need to find \( r \):
\[ r = \sqrt{\frac{154}{\pi}} \]
Then, substituting \( r \) into the height formula:
\[ h = 2\sqrt{\frac{154}{\pi}} \]

### Step 7: Calculate the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting \( h = 2r \):
\[ V = \pi r^2 (2r) = 2\pi r^3 \]
Now substituting \( r^2 = \frac{154}{\pi} \):
\[ V = 2\pi \left(\sqrt{\frac{154}{\pi}}\right)^3 = 2\pi \cdot \frac{154\sqrt{\frac{154}{\pi}}}{\pi^{3/2}} \]
Simplifying further:
\[ V = 2 \cdot \frac{154\sqrt{154}}{\pi^{1/2}} = \frac{308\sqrt{154}}{\sqrt{\pi}} \]

### Step 8: Numerical approximation
Using \( \pi \approx 3.14 \):
\[ \sqrt{\pi} \approx 1.77 \]
Calculating \( \sqrt{154} \approx 12.41 \):
\[ V \approx \frac{308 \cdot 12.41}{1.77} \approx 2156 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is approximately **2156 cm³**.