The cross-section of a tunnel is a square of side 7 m surmounted by a semicircle. Calculate: its floor area. The tunnel is 80 m long.

Answer

To solve the problem, we need to calculate the floor area of the tunnel, which has a cross-section that consists of a square and a semicircle. Here’s the step-by-step solution:

### Step 1: Understand the Cross-Section
The cross-section of the tunnel is a square with a side length of 7 m, surmounted by a semicircle. The square's dimensions are as follows:
- Each side of the square = 7 m

### Step 2: Calculate the Area of the Square
The area of the square can be calculated using the formula:
\[ \text{Area of the square} = \text{side}^2 \]
Substituting the side length:
\[ \text{Area of the square} = 7^2 = 49 \, \text{m}^2 \]

### Step 3: Calculate the Radius of the Semicircle
The semicircle is placed on top of the square. The diameter of the semicircle is equal to the side of the square. Therefore:
- Diameter of the semicircle = 7 m
- Radius of the semicircle = \(\frac{7}{2} = 3.5 \, \text{m}\)

### Step 4: Calculate the Area of the Semicircle
The area of a full circle is given by the formula:
\[ \text{Area of the circle} = \pi r^2 \]
Since we need the area of a semicircle, we will take half of the area of the circle:
\[ \text{Area of the semicircle} = \frac{1}{2} \pi r^2 \]
Substituting the radius:
\[ \text{Area of the semicircle} = \frac{1}{2} \pi (3.5)^2 = \frac{1}{2} \pi (12.25) \approx 19.63495 \, \text{m}^2 \]

### Step 5: Calculate the Total Cross-Sectional Area
Now, we add the area of the square and the area of the semicircle to find the total cross-sectional area:
\[ \text{Total cross-sectional area} = \text{Area of the square} + \text{Area of the semicircle} \]
\[ \text{Total cross-sectional area} = 49 + 19.63495 \approx 68.63495 \, \text{m}^2 \]

### Step 6: Calculate the Floor Area of the Tunnel
The floor area of the tunnel can be calculated by multiplying the total cross-sectional area by the length of the tunnel:
\[ \text{Floor area} = \text{Total cross-sectional area} \times \text{Length of the tunnel} \]
Given that the length of the tunnel is 80 m:
\[ \text{Floor area} = 68.63495 \times 80 \approx 5490.796 \, \text{m}^2 \]

### Final Answer
The floor area of the tunnel is approximately \(5490.80 \, \text{m}^2\).

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