A horse is tethered to one corner of a rectangular field of dimensions 70 m `xx ` 52 m, by a rope of length 21 m. How much area of the field can it graze?

Answer

To solve the problem of how much area a horse can graze when tethered to one corner of a rectangular field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
The horse is tethered at one corner of a rectangular field with dimensions 70 m by 52 m, and the length of the rope is 21 m. The area that the horse can graze will be a quarter of a circle (a quadrant) with a radius equal to the length of the rope.

2. **Identify the Radius**:
The radius \( r \) of the grazing area is equal to the length of the rope, which is 21 m.

3. **Calculate the Area of the Quadrant**:
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 of the circle (the quadrant), we will take one-fourth of the area of the full circle:
\[ \text{Area of the quadrant} = \frac{1}{4} \pi r^2 \]

4. **Substituting the Values**:
Substitute \( r = 21 \) m into the formula:
\[ \text{Area of the quadrant} = \frac{1}{4} \pi (21)^2 \]

5. **Calculating \( 21^2 \)**:
\[ 21^2 = 441 \]
So,
\[ \text{Area of the quadrant} = \frac{1}{4} \pi \times 441 \]

6. **Using \( \pi \approx \frac{22}{7} \)**:
\[ \text{Area of the quadrant} = \frac{1}{4} \times \frac{22}{7} \times 441 \]

7. **Calculating the Area**:
First, calculate \( \frac{441}{4} \):
\[ \frac{441}{4} = 110.25 \]
Now, multiply by \( \frac{22}{7} \):
\[ \text{Area} = 110.25 \times \frac{22}{7} \]
\[ = 110.25 \times 3.142857 \approx 346.5 \text{ m}^2 \]

8. **Final Answer**:
The area that the horse can graze is approximately \( 346.5 \text{ m}^2 \).

### Summary:
The area grazed by the horse is \( 346.5 \text{ m}^2 \).