A horse is tethered to one corner of a rectangular grassy field 40 m by 24 m with a rope 14 m long. Over how much area of the field can it graze?

Options

A. 154 `m^(2)`
B. 308 `m^(2)`
C. 150 `m^(2)`
D. None of these

Answer

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

### Step 1: Understand the Problem
The horse is tethered to one corner of a rectangular field measuring 40 m by 24 m with a rope that is 14 m long. The area the horse can graze will be limited by the length of the rope and the boundaries of the field.

### Step 2: Determine the Area of Grazing
Since the horse is tethered at one corner, the area it can graze will be a quarter of a circle with a radius equal to the length of the rope (14 m).

### Step 3: Calculate the Area of the Circle
The formula for the area of a circle is given by:
\[ \text{Area of Circle} = \pi r^2 \]
where \( r \) is the radius.

Substituting the radius:
\[ \text{Area of Circle} = \pi \times (14)^2 \]
Calculating \( 14^2 \):
\[ 14^2 = 196 \]
Thus,
\[ \text{Area of Circle} = \pi \times 196 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of Circle} = \frac{22}{7} \times 196 \]

### Step 4: Simplify the Calculation
To simplify:
\[ \frac{22}{7} \times 196 = \frac{22 \times 196}{7} \]
Calculating \( 196 \div 7 = 28 \):
\[ \text{Area of Circle} = 22 \times 28 = 616 \, \text{m}^2 \]

### Step 5: Calculate the Grazing Area
Since the horse can only graze in a quarter of the circle:
\[ \text{Area Grazed} = \frac{1}{4} \times \text{Area of Circle} = \frac{1}{4} \times 616 = 154 \, \text{m}^2 \]

### Final Answer
The area of the field that the horse can graze is:
\[ \text{Area Grazed} = 154 \, \text{m}^2 \]

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