The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 `cm^(2)`, find the volume of the cylinder. `(pi = (22)/(7))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the radius of the base of the cylinder be \( r \) cm and the height of the cylinder be \( h \) cm. According to the problem, we have:
\[ r + h = 37 \quad \text{(1)} \]

### Step 2: Write the formula for the total surface area (TSA) of the cylinder
The total surface area \( A \) of a right circular cylinder is given by the formula:
\[ A = 2\pi r^2 + 2\pi rh \]
We know from the problem that:
\[ A = 1628 \quad \text{(2)} \]

### Step 3: Substitute the value of \( h \) from equation (1) into equation (2)
From equation (1), we can express \( h \) in terms of \( r \):
\[ h = 37 - r \]
Now substitute \( h \) into the TSA formula:
\[ 1628 = 2\pi r^2 + 2\pi r(37 - r) \]
This simplifies to:
\[ 1628 = 2\pi r^2 + 74\pi r - 2\pi r^2 \]
\[ 1628 = 74\pi r \]

### Step 4: Solve for \( r \)
Now, we can substitute \( \pi = \frac{22}{7} \) into the equation:
\[ 1628 = 74 \times \frac{22}{7} r \]
To eliminate the fraction, multiply both sides by 7:
\[ 1628 \times 7 = 74 \times 22 r \]
\[ 11496 = 1628 r \]
Now, divide both sides by 1628:
\[ r = \frac{11496}{1628} \]
Calculating this gives:
\[ r = 7 \text{ cm} \]

### Step 5: Find the height \( h \)
Now, substitute \( r \) back into equation (1) to find \( h \):
\[ h = 37 - r = 37 - 7 = 30 \text{ cm} \]

### Step 6: 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 \( h \):
\[ V = \frac{22}{7} \times (7^2) \times 30 \]
\[ V = \frac{22}{7} \times 49 \times 30 \]
\[ V = \frac{22 \times 49 \times 30}{7} \]
Now, simplify:
\[ V = 22 \times 49 \times 30 \div 7 = 22 \times 210 = 4620 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is:
\[ V = 4620 \text{ cm}^3 \]