Four horses are tethered at four corners of a square plot of 42 m so that they just cannot reach one another . The area left un grazed is :

Options

A. `378 m^(2) `
B. `438 m^(2) `
C. `786 m^(2) `
D. none of these

Answer

To solve the problem of finding the area left ungrazed when four horses are tethered at the corners of a square plot of 42 m, we can follow these steps:

### Step 1: Calculate the area of the square plot.
The area \( A \) of a square is given by the formula:
\[ A = \text{side}^2 \]
Here, the side of the square plot is 42 m. Thus, the area is:
\[ A = 42^2 = 1764 \text{ m}^2 \]

### Step 2: Determine the grazing area for each horse.
Each horse is tethered at a corner of the square plot and can graze in a quarter-circle area. The radius of the grazing area is equal to the distance from the corner to the center of the square, which is half the side length of the square:
\[ \text{Radius} = \frac{42}{2} = 21 \text{ m} \]
The area \( A_h \) grazed by one horse is:
\[ A_h = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (21^2) = \frac{1}{4} \pi (441) = \frac{441\pi}{4} \text{ m}^2 \]

### Step 3: Calculate the total grazing area for all four horses.
Since there are four horses, the total grazing area \( A_g \) is:
\[ A_g = 4 \times A_h = 4 \times \frac{441\pi}{4} = 441\pi \text{ m}^2 \]

### Step 4: Calculate the ungrazed area.
The ungrazed area \( A_u \) is the area of the square plot minus the total grazing area:
\[ A_u = A - A_g = 1764 - 441\pi \]
Using \( \pi \approx \frac{22}{7} \):
\[ A_g \approx 441 \times \frac{22}{7} = 1386 \text{ m}^2 \]
Now substituting this back into the ungrazed area calculation:
\[ A_u = 1764 - 1386 = 378 \text{ m}^2 \]

### Conclusion
The area left ungrazed is:
\[ \boxed{378 \text{ m}^2} \]

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