Question

35 POINTS AVAILABLE

1.
In circle S, the area is 25pi cm². Find the length of each side of square SQUA.
Report your final answer with two decimal places.(see first pic below)
2.
Given that A is the center of the concentric circles and BCDE is a square with AB = 18, find the area of the shaded region. Report your final answer(see 2nd pic below)

Answer 1

Part 1) The length of each side of square AQUA is
`3.54\ cm`

Part 2) The area of the shaded region is
`(486\pi-648)\ units^(2)`

Explanation:

Part 1)

step 1

Find the radius of the circle S

The area of the circle is equal to

`A=\pi r^(2)`

we have

`A=25\pi\ cm^(2)`

substitute in the formula and solve for r

`25\pi=\pi r^(2)`

simplify

`25=r^(2)`

`r=5\ cm`

step 2

Find the length of each side of square SQUA

In the square SQUA

we have that

SQ=QU=UA=AS

`SU=r=5\ cm`

Let

x------> the length side of the square

Applying the Pythagoras Theorem

`5^(2)=x^(2) +x^(2)`

`5^(2)=2x^(2)`

`x^(2)=(25)/(2)\\ \\x=\sqrt{(25)/(2)}\ cm\\ \\ x=3.54\ cm`

Part 2) we know that

The area of the shaded region is equal to the area of the larger circle minus the area of the square plus the area of the smaller circle

Find the area of the larger circle

The area of the circle is equal to

`A=\pi r^(2)`

we have

`r=AB=18\ units`

substitute in the formula

`A=\pi (18)^(2)=324\pi\ units^(2)`

step 2

Find the length of each side of square BCDE

we have that

`AB=18\ units`

The diagonal DB is equal to

`DB=(2)18=36\ units`

Let

x------> the length side of the square BCDE

Applying the Pythagoras Theorem

`36^(2)=x^(2) +x^(2)`

`1,296=2x^(2)`

`648=x^(2)`

`x=√(648)\ units`

step 3

Find the area of the square BCDE

The area of the square is

`A=(√(648))^(2)=648\ units^(2)`

step 4

Find the area of the smaller circle

The area of the circle is equal to

`A=\pi r^(2)`

we have

`r=(√(648))/2\ units`

substitute in the formula

`A=\pi ((√(648))/2)^(2)=162\pi\ units^(2)`

step 5

Find the area of the shaded region

`324\pi\ units^(2)-648\ units^(2)+162\pi\ units^(2)=(486\pi-648)\ units^(2)`