A horse is placed for grazing inside a rectangular field 70 m by 52 m and is tethered to one corner by a rope 21 m long. On how much area can it graze?

Answer

To find the area that the horse can graze, we need to consider the shape of the area it can cover. Since the horse is tethered at one corner of the rectangular field with a rope of 21 meters, it can graze in the form of a quarter circle (quadrant) with a radius of 21 meters.

### Step-by-Step Solution:

1. **Identify the shape of the grazing area**:
The horse is tethered at one corner of the rectangular field, allowing it to graze in a quarter circle.

2. **Determine the radius of the grazing area**:
The length of the rope is given as 21 meters, which will be the radius (r) of the quarter circle.

3. **Calculate the area of the quarter circle**:
The formula for the area of a circle is:
\[ \text{Area of Circle} = \pi r^2 \]
Since we are dealing with a quarter circle, we will take one-fourth of the area of the circle:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \pi r^2 \]

4. **Substitute the radius into the formula**:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \times \pi \times (21)^2 \]

5. **Calculate \(21^2\)**:
\[ 21^2 = 441 \]

6. **Substitute \(441\) into the area formula**:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \times \pi \times 441 \]

7. **Use \(\pi \approx \frac{22}{7}\)** for calculation**:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \times \frac{22}{7} \times 441 \]

8. **Simplify the expression**:
\[ = \frac{22 \times 441}{4 \times 7} \]
\[ = \frac{22 \times 441}{28} \]

9. **Calculate \(22 \times 441\)**:
\[ 22 \times 441 = 9702 \]

10. **Now divide by 28**:
\[ \text{Area of Quarter Circle} = \frac{9702}{28} = 346.5 \text{ square meters} \]

### Final Answer:
The area that the horse can graze is **346.5 square meters**.