Question
A goat is tethered to a stake in the ground by a rope that is 10 ft. long. In terms of π, what is the maximum area the goat can graze in, with the stake at the center?
Responses
A 10pi;
10pi; ft 2
B 25π
25π ft 2
C 50π
50π ft 2
D 100π
Responses
A 10pi;
10pi; ft 2
B 25π
25π ft 2
C 50π
50π ft 2
D 100π
Answers
GPT-4o mini
To find the maximum area that the goat can graze, we need to calculate the area of a circle, since the goat can move around the stake in a circular pattern defined by the length of the rope.
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius of the circle. In this case, the radius \( r \) is equal to the length of the rope, which is 10 ft.
Now, we can substitute \( r \) into the area formula:
\[
A = \pi (10)^2 = \pi \times 100 = 100\pi
\]
Thus, the maximum area the goat can graze in is
\[
100\pi \text{ ft}^2.
\]
The correct response is:
**D) 100π**.
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius of the circle. In this case, the radius \( r \) is equal to the length of the rope, which is 10 ft.
Now, we can substitute \( r \) into the area formula:
\[
A = \pi (10)^2 = \pi \times 100 = 100\pi
\]
Thus, the maximum area the goat can graze in is
\[
100\pi \text{ ft}^2.
\]
The correct response is:
**D) 100π**.