Question

Anybody know the answers to these 3?

Answer 1

Answer:

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 $56.53\pi\ in^{2}$

Step-by-step 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

$\frac{9\pi }{360} =\frac{x }{84}\\ \\x=(9\pi)*84/360\\ \\x=2.1\pi\ m^{2}$

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

$\frac{10\pi }{360} =\frac{x }{90}\\ \\x=(10\pi)*90/360\\ \\x=2.5\pi\ in$

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

$\frac{64\pi }{360} =\frac{x }{318}\\ \\x=(64\pi)*318/360\\ \\x=56.53\pi\ in^{2}$