A right circular cylindrical tunnel of diameter 4 m and length 10 m is to be constructed from a sheet of iron . The area of the iron sheet required :

Options

A. `(280)/(pi)`
B. `40 pi `
C. `80 pi `
D. none of these

Answer

To find the area of the iron sheet required to construct a right circular cylindrical tunnel, we need to calculate the total surface area of the cylinder. Here are the steps to solve the problem:

### Step 1: Identify the dimensions of the cylinder
- **Diameter of the cylinder** = 4 m
- **Radius (r)** = Diameter / 2 = 4 m / 2 = 2 m
- **Length (height, h)** = 10 m

### Step 2: Use the formula for the total surface area of a cylinder
The total surface area (A) of a right circular cylinder is given by the formula:
\[ A = 2\pi r(h + r) \]
Where:
- \( r \) = radius of the cylinder
- \( h \) = height (length) of the cylinder

### Step 3: Substitute the values into the formula
Now, we substitute the values of \( r \) and \( h \) into the formula:
- \( r = 2 \) m
- \( h = 10 \) m

So,
\[ A = 2\pi(2)(10 + 2) \]

### Step 4: Simplify the expression
Calculate \( h + r \):
\[ h + r = 10 + 2 = 12 \]

Now substitute back into the area formula:
\[ A = 2\pi(2)(12) \]

### Step 5: Calculate the area
Now calculate:
\[ A = 2 \times 2 \times 12 \times \pi \]
\[ A = 48\pi \, \text{m}^2 \]

### Step 6: Final answer
The area of the iron sheet required to construct the cylindrical tunnel is:
\[ A \approx 48 \times 3.14 \, \text{m}^2 \approx 150.72 \, \text{m}^2 \]
Thus, the area of the iron sheet required is \( 48\pi \, \text{m}^2 \).

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