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

Answer:

2735.25  ft squared

Step-by-step 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 * $\frac{1}{2} ab$ 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) *$\frac{a}{2}$ = 3.63

=> area of the regular pentagon = 5*$\frac{1}{2} *10*3.63$ = 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