Find the total surface area of an open pipe of length 50 cm, external diameter 20 cm and internal diameter 6 cm.

Answer

To find the total surface area of an open pipe, we will follow these steps:

### Step 1: Identify the given values
- Length of the pipe (L) = 50 cm
- External diameter (D) = 20 cm
- Internal diameter (d) = 6 cm

### Step 2: Calculate the external radius (R)
The external radius (R) can be calculated using the formula:
\[ R = \frac{D}{2} \]
Substituting the value of D:
\[ R = \frac{20}{2} = 10 \text{ cm} \]

### Step 3: Calculate the internal radius (r)
The internal radius (r) can be calculated using the formula:
\[ r = \frac{d}{2} \]
Substituting the value of d:
\[ r = \frac{6}{2} = 3 \text{ cm} \]

### Step 4: Use the formula for the total surface area of the open pipe
The formula for the total surface area (TSA) of an open pipe is given by:
\[ \text{TSA} = 2\pi RL + 2\pi rL + 2\pi(R^2 - r^2) \]

### Step 5: Substitute the values into the formula
Substituting the values of R, r, and L into the formula:
\[ \text{TSA} = 2\pi(10)(50) + 2\pi(3)(50) + 2\pi(10^2 - 3^2) \]

### Step 6: Calculate each term
1. Calculate \( 2\pi(10)(50) \):
\[ 2 \times \frac{22}{7} \times 10 \times 50 = 1000\frac{22}{7} \]
2. Calculate \( 2\pi(3)(50) \):
\[ 2 \times \frac{22}{7} \times 3 \times 50 = 300\frac{22}{7} \]
3. Calculate \( 2\pi(10^2 - 3^2) \):
\[ 2\pi(100 - 9) = 2\pi(91) = 182\pi = 182 \times \frac{22}{7} \]

### Step 7: Combine all the terms
Now, we can combine all the terms:
\[ \text{TSA} = 1000\frac{22}{7} + 300\frac{22}{7} + 182\frac{22}{7} \]
\[ = \left(1000 + 300 + 182\right)\frac{22}{7} \]
\[ = 1482\frac{22}{7} \]

### Step 8: Calculate the final value
Now, we can calculate:
\[ \text{TSA} = \frac{1482 \times 22}{7} = \frac{32604}{7} \approx 4657.71 \text{ cm}^2 \]

Thus, the total surface area of the open pipe is approximately **4657.71 cm²**.

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