A horse is tethered to the corner of a rectangular field 50 m by 20 m by a 14 m long rope. What is the area of the field that it can graze?
Understand the Problem
The question asks us to calculate the area a horse can graze when tethered to the corner of a rectangular field, given the dimensions of the field and the length of the rope. Essentially, we need to find the area of a sector of a circle with a radius equal to the rope length.
Answer
$154 \text{ m}^2$
Answer for screen readers
(b) $154 \text{ m}^2$
Steps to Solve
- Identify the shape of the grazing area
Since the horse is tethered to a corner, it can graze in a quarter-circle (sector) with a radius equal to the length of the rope.
- Calculate the area of the full circle
The area of a full circle is given by the formula $A = \pi r^2$, where $r$ is the radius. In this case, $r = 14 \text{ m}$. Therefore, the area of the full circle would be: $A = \pi (14)^2 = 196\pi \text{ m}^2$
- Calculate the area of the quarter-circle
Since the horse is tethered to the corner of a rectangular field, it can only graze a quarter of the circle. Therefore, divide the full circle area by 4:
$\text{Area of quarter-circle} = \frac{196\pi}{4} = 49\pi \text{ m}^2$
- Approximate $\pi$ and calculate the final area
Using $\pi \approx \frac{22}{7}$, we have:
$\text{Area} = 49 \times \frac{22}{7} = 7 \times 22 = 154 \text{ m}^2$
(b) $154 \text{ m}^2$
More Information
The area the horse can graze is $154 \text{ m}^2$. This solution assumes that the rope is long enough that the horse is not restricted by the sides of the rectangular field; that is, the radius of the quarter circle is 14 m.
Tips
A common mistake would be to calculate the area of the full circle instead of the quarter-circle. Always consider the geometry of the problem.
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