The sum of the radius and the height of a cyclinder is 37 cm and the total surface area of the cyclinder is ` 1628 cm ^(2)`. Find the height and the volume of the cyclinder.

Answer

To solve the problem step by step, we will follow the information given and use the formulas for the surface area and volume of a cylinder.

### Step 1: Set up the equations
We are given two pieces of information:
1. The sum of the radius (r) and the height (h) of the cylinder is 37 cm:
\[ r + h = 37 \quad \text{(1)} \]
2. The total surface area (TSA) of the cylinder is 1628 cm². The formula for the total surface area of a cylinder is:
\[ \text{TSA} = 2\pi r (r + h) \quad \text{(2)} \]

### Step 2: 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 this value of \( h \) into the TSA equation (2):
\[ 2\pi r (r + (37 - r)) = 1628 \]
This simplifies to:
\[ 2\pi r \cdot 37 = 1628 \]

### Step 3: Solve for \( r \)
Now, we can simplify the equation:
\[ 74\pi r = 1628 \]
Dividing both sides by \( 74\pi \):
\[ r = \frac{1628}{74\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{1628 \times 7}{74 \times 22} \]
Calculating this gives:
\[ r = \frac{11496}{1628} = 7 \, \text{cm} \]

### Step 4: Find the height \( h \)
Now that we have \( r \), we can find \( h \) using equation (1):
\[ h = 37 - r = 37 - 7 = 30 \, \text{cm} \]

### Step 5: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi (7^2) (30) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 49 \times 30 \]
Calculating this gives:
\[ V = \frac{22 \times 1470}{7} = 4620 \, \text{cm}^3 \]

### Final Answers
- Height \( h = 30 \, \text{cm} \)
- Volume \( V = 4620 \, \text{cm}^3 \)