To find the area that the cow can graze when it is tied to a pole at the midpoint of one side of a square field whose dimensions are 40 meters by 40 meters, we can follow these steps:
1. Understand the setup:
- The square field has each side measuring 40 meters.
- The pole to which the cow is tied is located at the midpoint of one side of the square.
- The cow is tied with a 14-meter-long rope.
2. Area of a quarter circle:
- The cow can move in a quarter-circle because it is tied at the midpoint of one of the sides of the square.
- The radius of this quarter circle is 14 meters, which is the length of the rope.
3. Formula for the area of a quarter circle:
- The formula for the area of a full circle is [tex]\( \pi \times r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- Since the cow can only graze a quarter of the circle, we need to take one-fourth of the full circle's area.
- Hence, the area of the quarter circle is [tex]\( \frac{1}{4} \times \pi \times r^2 \)[/tex].
4. Substitute the radius into the formula:
- Here, [tex]\( r = 14 \)[/tex] meters.
- So, the area of the quarter circle grazed by the cow is [tex]\( \frac{1}{4} \times \pi \times 14^2 \)[/tex].
5. Calculate the area:
- First, calculate [tex]\( 14^2 \)[/tex]:
[tex]\[
14^2 = 196
\][/tex]
- Then, multiply by [tex]\( \pi \)[/tex] (approximately [tex]\( 3.14159 \)[/tex]):
[tex]\[
\pi \times 196 \approx 3.14159 \times 196 \approx 615.75216
\][/tex]
- Now, take one-fourth of this area to find the area of the quarter circle:
[tex]\[
\frac{1}{4} \times 615.75216 \approx 153.93804
\][/tex]
Therefore, the area that the cow can graze is approximately [tex]\( 153.93804 \)[/tex] square meters.
