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 describes a scenario where a horse is tethered to the corner of a rectangular field and asks for the area the horse can graze. It is asking us to calculate the area of a sector of a circle with a radius equal to the length of the rope.
Answer
The area the horse can graze is $154$ m$^2$.
Answer for screen readers
(b) $154$ m$^2$
Steps to Solve
- Identify the shape of the grazing area
Since the horse is tethered to a corner of a rectangular field, it can graze in a quarter-circle shape. This is a sector of a circle with a central angle of 90 degrees (or $\pi/2$ radians).
- Determine the radius of the quarter-circle
The radius of the quarter-circle is the length of the rope, which is given as 14 m.
- Calculate the area of the full circle
The area of a full circle with radius $r$ is $A = \pi r^2$. In this case, $r = 14$ m, so the area is $A = \pi (14^2) = 196\pi$ m$^2$.
- Calculate the area of the quarter-circle
Since the horse can graze in a quarter-circle, we need to find one-fourth of the area of the full circle:
$A_{\text{quarter-circle}} = \frac{1}{4} \pi r^2 = \frac{1}{4} \pi (14^2) = \frac{1}{4} (196\pi) = 49\pi$
- Approximate the value using $\pi \approx \frac{22}{7}$
$A_{\text{quarter-circle}} = 49 \times \frac{22}{7} = 7 \times 22 = 154$ m$^2$
(b) $154$ m$^2$
More Information
The area that the horse can graze is a quarter-circle because it is tethered to the corner of a rectangle, allowing it to graze over 90 degrees.
Tips
A common mistake would be calculating the area of the full circle instead of the quarter-circle. Another mistake could be using the wrong radius. Also not remembering the formula for the area of a circle.
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