The total surface area of a solid right circular cylinder of height 13cm, is 880 `cm^(2)` . Its volume (in `cm^(2)`) is 11k. The value of k is:
(Taken `pi=22/7`)

Options

A. 208
B. 182
C. 104
D. 91

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the formulas
The total surface area (TSA) of a right circular cylinder is given by the formula:
\[ \text{TSA} = 2\pi r(h + r) \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Substitute the known values
We know the total surface area is 880 cm² and the height \( h \) is 13 cm. Substituting these values into the TSA formula:
\[ 880 = 2 \times \frac{22}{7} \times r \times (13 + r) \]

### Step 3: Simplify the equation
First, simplify the equation:
\[ 880 = \frac{44}{7} r (13 + r) \]
Multiplying both sides by 7 to eliminate the fraction:
\[ 6160 = 44r(13 + r) \]

### Step 4: Divide by 44
Now, divide both sides by 44:
\[ 140 = r(13 + r) \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ r^2 + 13r - 140 = 0 \]

### Step 6: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ (r - 7)(r + 20) = 0 \]
This gives us two possible solutions for \( r \):
1. \( r = 7 \)
2. \( r = -20 \) (not valid since radius cannot be negative)

Thus, the radius \( r \) is 7 cm.

### Step 7: Calculate the volume
Now we can calculate the volume \( V \) of the cylinder using the formula:
\[ V = \pi r^2 h \]
Substituting \( r = 7 \) cm and \( h = 13 \) cm:
\[ V = \frac{22}{7} \times 7^2 \times 13 \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now substituting back:
\[ V = \frac{22}{7} \times 49 \times 13 \]

### Step 8: Simplify the volume calculation
Calculating \( \frac{22 \times 49 \times 13}{7} \):
\[ V = 22 \times 7 \times 13 \]
Calculating \( 22 \times 7 \):
\[ 22 \times 7 = 154 \]
Now calculating \( 154 \times 13 \):
\[ 154 \times 13 = 2002 \]

### Step 9: Relate volume to k
We are given that the volume is \( 11k \):
\[ 11k = 2002 \]
Now, solving for \( k \):
\[ k = \frac{2002}{11} = 182 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{182} \]