Question

A goat is tied at one corner of a 12 feet by 16 feet barn. If the rope is 24 feet long, find the area of the region outside the barn, in which the goats can graze.

Answer 1

1520.5 ft²

Explanation:

A diagram can be helpful.

The total area is the sum of ...

3/4 of a circle with radius 24 ft

1/4 of a circle with radius 12 ft

1/4 of a circle with radius 8 feet

__

The area of a whole circle is given by ...

A = πr²

so our grazing area is ...

A = (3/4)π(24 ft)² + (1/4)π(12 ft)² + (1/4)π(8 ft)²

= π(432 ft² +36 ft² +16 ft²) = 484π ft²

A ≈ 1520.5 ft²