Tiana would like to buy cone-shaped hats for her birthday partyShe wants hats with the largest volume, since they look bigger. Which of the following brands of hat should she buy?

The area of the shaded portion of the figure is

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The shaded area is equal to the area of the square less the area not shaded.

There are 4 "not shaded" regions.

step 1

Find the area of square ABCD

The area of square is equal to

where

b is the length side of the square

we have

substitute

step 2

We can find the area of 2 "not shaded" regions by calculating the area of the square less two semi-circles (one circle):

The area of circle is equal to

The diameter of the circle is equal to the length side of the square

so

---> radius is half the diameter

substitute

Therefore, the area of 2 "not-shaded" regions is:

and the area of 4 "not-shaded" regions is:

step 3

Find the area of the shaded region

Remember that the area of the shaded region is the area of the square less 4 "not shaded" regions:

so

---> exact value

assume

substitute