A horse is tied to a pole fixed at one corner of a 14 m `xx` 14 m square field of grass by means of a rope 7 m long . Find the area of the square field within which the horse can graze .

Options

A. `77 m^(2)`
B. `196 m^(2)`
C. `28 m^(2)`
D. `38.5 m^(2)`

Answer

To solve the problem of finding the area of the square field within which the horse can graze, we can follow these steps:

### Step 1: Understand the Problem
The horse is tied to a pole at one corner of a square field measuring 14 m x 14 m with a rope that is 7 m long. The horse can graze in a circular area defined by the length of the rope.

**Hint:** Visualize the square field and the circular area that the horse can graze.

### Step 2: Determine the Radius of Grazing
The length of the rope is 7 m, which means the radius (r) of the circular area where the horse can graze is 7 m.

**Hint:** Remember that the radius is the distance from the center of the circle (the pole) to the edge of the circle.

### Step 3: Calculate the Area of the Circle
The area (A) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (7)^2 = \pi \times 49 \]

**Hint:** Use the approximate value of \(\pi\) as \( \frac{22}{7} \) for calculations.

### Step 4: Calculate the Area of the Circle
Now, we substitute \(\pi\) with \(\frac{22}{7}\):
\[ A = \frac{22}{7} \times 49 \]

### Step 5: Simplify the Calculation
Calculating the area:
\[ A = \frac{22 \times 49}{7} \]
\[ A = \frac{1078}{7} \]
\[ A = 154 \text{ m}^2 \]

### Step 6: Find the Area Accessible to the Horse
Since the horse is tied at one corner of the square field, it can only graze in one-quarter of the circle:
\[ \text{Area accessible to the horse} = \frac{1}{4} \times 154 \]
\[ = 38.5 \text{ m}^2 \]

### Final Answer
The area of the square field within which the horse can graze is **38.5 m²**.

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