A horse is tied to a pole fixed at one corner of a `50 m xx 50 m` square field of grass by means of a 20 m long rope. What is the area to the nearest whole number of that part of the field which the horse can graze?

Options

A. `1256 m^2`
B. `942 m^2 `
C. `628 m^2`
D. `314 m^2`

Answer

To solve the problem of finding the area of the part of the field that the horse can graze, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
The horse is tied to a pole at one corner of a square field measuring 50 m x 50 m with a rope that is 20 m long. The horse can graze in a circular area defined by the length of the rope, but since it is tied at a corner, it can only graze in a quarter circle.

**Hint**: Visualize the square field and the position of the horse tied at the corner.

2. **Identify the Radius**:
The radius of the grazing area is equal to the length of the rope, which is 20 m.

**Hint**: Remember that the radius is the distance from the pole to the edge of the grazing area.

3. **Calculate the Area of the Quarter Circle**:
The area \( A \) of a full circle is given by the formula:
\[ A = \pi r^2 \]
Since the horse can only graze in a quarter circle, we need to calculate the area of a quarter circle:
\[ A_{quarter} = \frac{1}{4} \times \pi r^2 \]

4. **Substitute the Values**:
Here, \( r = 20 \) m. Therefore:
\[ A_{quarter} = \frac{1}{4} \times \pi \times (20)^2 \]
\[ A_{quarter} = \frac{1}{4} \times \pi \times 400 \]
\[ A_{quarter} = \frac{400\pi}{4} = 100\pi \]

5. **Use the Approximation for \(\pi\)**:
We can use \( \pi \approx \frac{22}{7} \) for our calculations:
\[ A_{quarter} = 100 \times \frac{22}{7} \]
\[ A_{quarter} = \frac{2200}{7} \approx 314.29 \text{ m}^2 \]

6. **Round to the Nearest Whole Number**:
The area to the nearest whole number is:
\[ A_{quarter} \approx 314 \text{ m}^2 \]

### Final Answer:
The area of the part of the field which the horse can graze is approximately **314 m²**.

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