Question

Anybody know the answers to these 3?

Answer 1

Part 1) The area of the shaded region is
`2.1\pi\ m^(2)`

Part 2) The length of the arc AB is
`2.5\pi\ in`

Part 3) The area of the shaded region is
`56.53\pi\ in^(2)`

Explanation:

Part 1) Find the area of the shaded region

step 1

Find the area of the circle

The area is equal to

`A=\pi r^(2)`

we have

`r=3\ m`

substitute

`A=\pi (3)^(2)`

`A=9\pi\ m^(2)`

step 2

we know that

The area of complete circle subtends a central angle of 360 degrees

so

by proportion

calculate the area of the shaded region with a central angle of 84 degrees

`(9\pi )/(360) =(x )/(84)\\ \\x=(9\pi)*84/360\\ \\x=2.1\pi\ m^(2)`

Part 2) What is the length of arc AB?

step 1

we know that

The circumference of a circle is equal to

`C=2\pi r`

we have

`r=5\ in`

substitute

`C=2\pi (5)`

`C=10\pi\ in`

step 2

we know that

The length of complete circle subtends a central angle of 360 degrees

so

by proportion

calculate the length of the arc AB with a central angle of 90 degrees

`(10\pi )/(360) =(x )/(90)\\ \\x=(10\pi)*90/360\\ \\x=2.5\pi\ in`

Part 3) Find the area of the shaded region given that XY measures 8 in

step 1

Find the area of the circle

The area is equal to

`A=\pi r^(2)`

we have

`XY=r=8\ in`

substitute

`A=\pi (8)^(2)`

`A=64\pi\ in^(2)`

step 2

we know that

The area of complete circle subtends a central angle of 360 degrees

so

by proportion

calculate the area of the shaded region with a central angle of (360-42)=318 degrees

`(64\pi )/(360) =(x )/(318)\\ \\x=(64\pi)*318/360\\ \\x=56.53\pi\ in^(2)`