Question

Your cousin has a goat in his backyard, tied with a 4 foot rope. The goat can move in a circular motion the length of the rope. What is the area of the backyard where his goat can eat the grass? Hint: The 4 foot rope represents the radius of the circle.
1. What equation can be used to calculate the area of the yard that his goat can eat the grass?
2. What is the area of this circle in your cousin’s backyard?
3. What would the new area be if he decides to use a 5 foot rope?

Answer 1

The stake the rope is tied to is the center of the circle. If the goat pulls the rope as far as it can go and walks around like this he makes a circle and he can eat the grass anywhere within this circle.

(a) Since the area of a circle is given by $A= \pi r^{2}$ this is the formula we can use to find the area of the yard where the goat can roam/eat/reach.

(b) Since the rope is 4 feet the radius r = 4 and the area is given by $4^{2} \pi =16 \pi ^{2}$ square feet. That's the exact area. For an approximation you could substitute 3.14 for pi and get 50.24 square feet.

(c)Since the rope is 5 feet the radius r = 4 and the area is given by $5^{2} \pi =25 \pi ^{2}$ square feet. That's the exact area. For an approximation you could substitute 3.14 for pi and get 78.5 square feet.