Question

Three semicircles are placed in a rectangle, as shown below. The length of the rectangle is 30 m . Find the area of the shaded region. Use the value for , and do not round your answer. Be sure to include the correct unit in your answer.

Answer 1

The area of the shaded region is: 32.25m^2

Explanation:

Given

See attachment for rectangle and semicircles

Required

The shaded region

We have:

`l =30` --- length of the rectangle

This means that the diameters of the three circles add up to 30m.

So, the diameter of 1 is:

`d = l/3`

`d = 30/3`

`d = 10`

The area of one circle is:

`A_1 = \pi * (d^2)/(8)`

`A_1 = 3.14 * (10^2)/(8)`

`A_1 = 3.14 * (100)/(8)`

`A_1 = 39.25m^2`

The area of the three circles is:

`A_2 = 3 * A_1`

`A_2 = 3 *39.25m^2`

`A_2 = 117.75m^2`

The area of the rectangle is:

`A_3 = 30 * r`

Where r is the radius of the circle

And

`r =d/2`

`r =10/2`

`r =5`

So, we have:

`A_3 = 30 * r`

`A_3 = 30 * 5`

`A_3 = 150`

So, the shaded region is:

`A_4 = A_3 - A_2`

`A_4 = 150 - 117.75`

`A_4 = 32.25m^2`