The sum of the height and the radius of a solid cylinder is 35 cm and its total surface area is `3080 cm^(2),` find the volume of the cylinder.

Answer

To solve the problem, we will follow these steps:

### Step 1: Set up the equations
Let the radius of the cylinder be \( r \) cm and the height be \( h \) cm. According to the problem, we have two equations:
1. \( r + h = 35 \) (the sum of the height and radius)
2. The total surface area of the cylinder is given by the formula:
\[ \text{Total Surface Area} = 2\pi r^2 + 2\pi rh \]
Given that the total surface area is \( 3080 \, \text{cm}^2 \), we can write:
\[ 2\pi r^2 + 2\pi rh = 3080 \]

### Step 2: Substitute \( h \) in terms of \( r \)
From the first equation \( r + h = 35 \), we can express \( h \) as:
\[ h = 35 - r \]

### Step 3: Substitute \( h \) in the surface area equation
Now, substitute \( h \) in the total surface area equation:
\[ 2\pi r^2 + 2\pi r(35 - r) = 3080 \]
Expanding this gives:
\[ 2\pi r^2 + 70\pi r - 2\pi r^2 = 3080 \]
This simplifies to:
\[ 70\pi r = 3080 \]

### Step 4: Solve for \( r \)
Now, divide both sides by \( 70\pi \):
\[ r = \frac{3080}{70\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{3080 \times 7}{70 \times 22} = \frac{21656}{1540} = 14 \, \text{cm} \]

### Step 5: Find \( h \)
Now that we have \( r \), we can find \( h \):
\[ h = 35 - r = 35 - 14 = 21 \, \text{cm} \]

### Step 6: Calculate the volume of the cylinder
The volume \( V \) of the cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi (14)^2 (21) \]
Calculating this gives:
\[ V = \pi \times 196 \times 21 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 196 \times 21 \]
Calculating \( 196 \times 21 = 4116 \):
\[ V = \frac{22 \times 4116}{7} = \frac{90552}{7} = 12936 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is \( 12936 \, \text{cm}^3 \).

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