The sum of the radius of the base of a solid cylinder and the height of the cylinder is 15 cm . If the total surface area of the cylinder is `660 cm^(2)` , then find the volume of the cylinder .

Options

A. `1232 cm^(3)`
B. `1256 cm^(3)`
C. `1296 cm^(3)`
D. `1276 cm^(3)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the given information
We know:
- The sum of the radius (r) and the height (h) of the cylinder is 15 cm:
\[ r + h = 15 \quad \text{(1)} \]
- The total surface area (TSA) of the cylinder is 660 cm²:
\[ \text{TSA} = 2\pi r (r + h) = 660 \quad \text{(2)} \]

### Step 2: Substitute the value of (r + h) in the TSA formula
From equation (1), we can substitute \( r + h \) in equation (2):
\[ \text{TSA} = 2\pi r (15) = 660 \]
This simplifies to:
\[ 30\pi r = 660 \]

### Step 3: Solve for r
Now, divide both sides by 30:
\[ \pi r = \frac{660}{30} = 22 \]
Now, divide both sides by \(\pi\):
\[ r = \frac{22}{\pi} \]
Using \(\pi \approx 3.14\) or \(\frac{22}{7}\), we can calculate \(r\):
\[ r \approx \frac{22}{3.14} \approx 7 \text{ cm} \]

### Step 4: Find the height (h)
Now that we have \(r\), we can find \(h\) using equation (1):
\[ r + h = 15 \implies 7 + h = 15 \]
So,
\[ h = 15 - 7 = 8 \text{ cm} \]

### Step 5: Calculate the volume of the cylinder
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ V = \pi (7^2)(8) = \pi (49)(8) = 392\pi \]
Using \(\pi \approx 3.14\):
\[ V \approx 392 \times 3.14 \approx 1231.68 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is approximately \(1231.68 \text{ cm}^3\).

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