There is a vast grassy farm in which there is a rectangular building of the farm-house whose length and breadth is 50 m and 40m respectively . A horse is tethered at a corner of the house with a tether of 80 m long. What is the maximum area that the horse can graze ?

Options

A. a. `5425 pi `
B. b. `5245 pi`
C. c. `254 pi`
D. d. none of these

Answer

To find the maximum area that the horse can graze, we need to consider the tether length and the position of the house. The horse is tethered at a corner of a rectangular building with a length of 50 m and a breadth of 40 m. The tether length is 80 m.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- The horse is tethered at a corner of the rectangular farmhouse.
- The dimensions of the farmhouse are:
- Length (L) = 50 m
- Breadth (B) = 40 m
- The length of the tether (r) = 80 m.

2. **Visualizing the Grazing Area**:
- When tied at the corner, the horse can graze in a circular area with a radius equal to the length of the tether, which is 80 m.
- However, the farmhouse will obstruct part of this circular area.

3. **Calculating the Grazing Area**:
- The horse can graze in a sector of a circle with a radius of 80 m. The angle of this sector is 270 degrees (as it can graze around the corner).
- The area of a full circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
- The area of the sector grazed by the horse is:
\[ \text{Area}_{\text{sector}} = \frac{270}{360} \times \pi \times (80)^2 \]
- Simplifying this:
\[ \text{Area}_{\text{sector}} = \frac{3}{4} \times \pi \times 6400 = 4800\pi \text{ m}^2 \]

4. **Adding Additional Grazing Areas**:
- After grazing around the corner, the horse can also graze in two additional quarter-circle areas:
- One with a radius of 30 m (the remaining distance after the farmhouse's breadth of 40 m).
- One with a radius of 40 m (the remaining distance after the farmhouse's length of 50 m).
- The area for the 30 m radius quarter-circle:
\[ \text{Area}_{30m} = \frac{1}{4} \times \pi \times (30)^2 = \frac{1}{4} \times \pi \times 900 = 225\pi \text{ m}^2 \]
- The area for the 40 m radius quarter-circle:
\[ \text{Area}_{40m} = \frac{1}{4} \times \pi \times (40)^2 = \frac{1}{4} \times \pi \times 1600 = 400\pi \text{ m}^2 \]

5. **Total Grazing Area**:
- Now, we sum all the areas:
\[ \text{Total Area} = 4800\pi + 225\pi + 400\pi = (4800 + 225 + 400)\pi = 4925\pi \text{ m}^2 \]
- Approximating \(\pi\) as 3.14:
\[ \text{Total Area} \approx 4925 \times 3.14 \approx 15460.5 \text{ m}^2 \]

### Final Answer:
The maximum area that the horse can graze is approximately \(15460.5 \text{ m}^2\).