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

Answer

To solve the problem, we need to find the volume of a solid right circular cylinder given the sum of the radius and height, and the total surface area.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the radius of the base of the cylinder be \( r \) cm and the height be \( h \) cm. According to the problem, we have:
\[ r + h = 37 \quad (1) \]

2. **Total Surface Area Formula:**
The total surface area (TSA) of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
We know from the problem that the TSA is 1628 sq.cm. Substituting this into the formula gives:
\[ 2\pi r (r + h) = 1628 \quad (2) \]

3. **Substituting the Value of \( r + h \):**
From equation (1), we can substitute \( r + h \) in equation (2):
\[ 2\pi r \cdot 37 = 1628 \]
Now substituting \( \pi = \frac{22}{7} \):
\[ 2 \cdot \frac{22}{7} \cdot r \cdot 37 = 1628 \]

4. **Solving for \( r \):**
Simplifying the equation:
\[ \frac{44}{7} \cdot 37r = 1628 \]
\[ 44 \cdot 37r = 1628 \cdot 7 \]
\[ 1628 \cdot 7 = 11396 \]
\[ 44 \cdot 37r = 11396 \]
\[ 1628r = \frac{11396}{44} \]
\[ r = \frac{11396}{44 \cdot 37} \]
\[ r = \frac{11396}{1628} = 7 \]

5. **Finding the Height \( h \):**
Now that we have \( r = 7 \) cm, we can find \( h \) using equation (1):
\[ r + h = 37 \]
\[ 7 + h = 37 \]
\[ h = 37 - 7 = 30 \text{ cm} \]

6. **Calculating 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 \( h \):
\[ V = \frac{22}{7} \cdot (7^2) \cdot 30 \]
\[ V = \frac{22}{7} \cdot 49 \cdot 30 \]
\[ V = \frac{22 \cdot 49 \cdot 30}{7} \]
\[ V = 22 \cdot 49 \cdot 30 / 7 = 22 \cdot 210 = 4620 \text{ cubic cm} \]

### Final Answer:
The volume of the cylinder is \( 4620 \, \text{cm}^3 \).

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