A horse is tethered to a pole with a 14 metre long rope at the corner of a 40 metre long and 24 metre wide rectangular grass-field. What area of the field will the horse graze?

Options

A. `154m^2`
B. `308 m^2`
C. `240m^2`
D. `480m^2`

Explanation

To solve the problem of how much area the horse can graze in the rectangular field, we can follow these steps:

### Step 1: Understand the Setup
The horse is tethered to a pole at one corner of a rectangular field that measures 40 meters in length and 24 meters in width. The rope is 14 meters long.

### Step 2: Identify the Grazing Area
Since the horse is tied at a corner, it can graze in a quarter-circle area with a radius equal to the length of the rope (14 meters).

### Step 3: Calculate the Area of the Quarter Circle
The formula for the area of a circle is:
\[ \text{Area} = \pi r^2 \]
Since the horse can only graze a quarter of the circle, we will use:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \pi r^2 \]
Substituting the radius \( r = 14 \) meters:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \pi (14)^2 \]

### Step 4: Calculate \( (14)^2 \)
Calculating \( (14)^2 \):
\[ (14)^2 = 196 \]

### Step 5: Substitute and Calculate the Area
Now substituting back into the area formula:
\[ \text{Area of Quarter Circle} = \frac{1}{4} \pi (196) \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of Quarter Circle} = \frac{1}{4} \times \frac{22}{7} \times 196 \]

### Step 6: Simplify the Calculation
Calculating:
\[ \frac{1}{4} \times 196 = 49 \]
Now substituting:
\[ \text{Area of Quarter Circle} = \frac{22}{7} \times 49 \]
Calculating:
\[ \frac{22 \times 49}{7} = 22 \times 7 = 154 \text{ square meters} \]

### Final Answer
Thus, the area of the field that the horse will graze is **154 square meters**.

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