If the sum of the radius and the height of a closed cylinder is 35 cm and the total surface area of the cylinder is `1540 cm^(2)`, Then the circumference of the base of the cylinder is :

Options

A. 66 cm
B. 44 cm
C. 56 cm
D. can't be determined

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the variables
Let the radius of the cylinder be \( r \) cm and the height be \( h \) cm. According to the problem, we have:
\[ r + h = 35 \, \text{cm} \]

### Step 2: Use the formula for the total surface area of a closed cylinder
The formula for the total surface area (TSA) of a closed cylinder is given by:
\[ \text{TSA} = 2\pi r (r + h) \]
We know that the total surface area is \( 1540 \, \text{cm}^2 \), so we can set up the equation:
\[ 2\pi r (r + h) = 1540 \]

### Step 3: Substitute \( r + h \) into the TSA equation
Since we know \( r + h = 35 \), we can substitute this into the TSA equation:
\[ 2\pi r (35) = 1540 \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 70\pi r = 1540 \]

### Step 5: Solve for \( r \)
To isolate \( r \), divide both sides by \( 70\pi \):
\[ r = \frac{1540}{70\pi} \]

### Step 6: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{1540}{70 \times \frac{22}{7}} \]
\[ r = \frac{1540 \times 7}{70 \times 22} \]

### Step 7: Calculate \( r \)
Now, simplify the expression:
\[ r = \frac{10780}{1540} \]
\[ r = 7 \, \text{cm} \]

### Step 8: Find the height \( h \)
Using \( r + h = 35 \):
\[ 7 + h = 35 \]
\[ h = 35 - 7 = 28 \, \text{cm} \]

### Step 9: Calculate the circumference of the base
The circumference \( C \) of the base of the cylinder is given by:
\[ C = 2\pi r \]
Substituting \( r = 7 \):
\[ C = 2 \times \frac{22}{7} \times 7 \]
\[ C = 2 \times 22 = 44 \, \text{cm} \]

### Final Answer
The circumference of the base of the cylinder is \( 44 \, \text{cm} \).

---