Question

a dog is tied to a 30-foot leash that is tied to the corner of a building in the shape of a regular pentagon with a 10-foot sides. how much area does the dog have in which to play?​

Answer 1

2735.25 ft squared

Explanation:

If a dog is tied to a 30-foot leash that is tied to the corner of a building, we can image that the corner is the center of a circle that has a radius of 30-foot (same length with the leash).

So the area of the circle is:

Area = pi R² = 3.14*
`30^(2)` = 2826 ft squared

Sadly, he's not out in the middle of a field. He's tied to the corner of the building so he can't cover the whole circle, because the building blocks a regular pentagon with a 10-foot sides.

We need to find out the area of the regular pentagon

A = 5 *
`(1)/(2) ab` where a is the base and b is the height.

with the property in a regular pentagon, we know that the height if it is:

b = tan(36) *
`(a)/(2)` = 3.63

=> area of the regular pentagon = 5*
`(1)/(2) *10*3.63` = 90.75 ft squared

=> area the dog have in which to play = area of the circle - area of the regular pentagon = 2826 - 90.75 = 2735.25 ft squared