The height of a hollow cylinder is 14 cm. If external diameter is 16 cm and total curved surface area of the hollow cylinder is 1320 sq. cm, then its internal diameter is

Options

A. 14 cm
B. 16 cm
C. 7 cm
D. 8 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know the following:
- Height of the hollow cylinder (h) = 14 cm
- External diameter (D) = 16 cm
- Total curved surface area (TSA) = 1320 cm²

### Step 2: Calculate the external radius
The external radius (R) can be calculated from the external diameter:
\[ R = \frac{D}{2} = \frac{16 \, \text{cm}}{2} = 8 \, \text{cm} \]

### Step 3: Write the formula for Total Curved Surface Area of a hollow cylinder
The formula for the total curved surface area (TSA) of a hollow cylinder is given by:
\[ \text{TSA} = 2\pi h(R + r) \]
where \( R \) is the external radius and \( r \) is the internal radius.

### Step 4: Substitute the known values into the TSA formula
We can substitute the known values into the TSA formula:
\[ 1320 = 2 \times \frac{22}{7} \times 14 \times (8 + r) \]

### Step 5: Simplify the equation
First, calculate \( 2 \times \frac{22}{7} \times 14 \):
\[ 2 \times \frac{22}{7} \times 14 = \frac{44 \times 14}{7} = \frac{616}{7} \]
Now, substitute this back into the equation:
\[ 1320 = \frac{616}{7} \times (8 + r) \]

### Step 6: Multiply both sides by 7 to eliminate the fraction
\[ 1320 \times 7 = 616 \times (8 + r) \]
\[ 9240 = 616 \times (8 + r) \]

### Step 7: Divide both sides by 616
\[ \frac{9240}{616} = 8 + r \]
Calculating \( \frac{9240}{616} \):
\[ \frac{9240}{616} = 15 \]
So,
\[ 15 = 8 + r \]

### Step 8: Solve for the internal radius
Now, isolate \( r \):
\[ r = 15 - 8 = 7 \, \text{cm} \]

### Step 9: Calculate the internal diameter
The internal diameter (d) is given by:
\[ d = 2r = 2 \times 7 = 14 \, \text{cm} \]

### Final Answer
The internal diameter of the hollow cylinder is **14 cm**.

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