Answer:
Part 1) The area of the shaded region is [tex]2.1\pi\ m^{2}[/tex]
Part 2) The length of the arc AB is [tex]2.5\pi\ in[/tex]
Part 3) The area of the shaded region is [tex]56.53\pi\ in^{2}[/tex]
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
[tex]A=\pi r^{2}[/tex]
we have
[tex]r=3\ m[/tex]
substitute
[tex]A=\pi (3)^{2}[/tex]
[tex]A=9\pi\ m^{2}[/tex]
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
[tex]\frac{9\pi }{360} =\frac{x }{84}\\ \\x=(9\pi)*84/360\\ \\x=2.1\pi\ m^{2}[/tex]
Part 2) What is the length of arc AB?
step 1
we know that
The circumference of a circle is equal to
[tex]C=2\pi r[/tex]
we have
[tex]r=5\ in[/tex]
substitute
[tex]C=2\pi (5)[/tex]
[tex]C=10\pi\ in[/tex]
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
[tex]\frac{10\pi }{360} =\frac{x }{90}\\ \\x=(10\pi)*90/360\\ \\x=2.5\pi\ in[/tex]
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
[tex]A=\pi r^{2}[/tex]
we have
[tex]XY=r=8\ in[/tex]
substitute
[tex]A=\pi (8)^{2}[/tex]
[tex]A=64\pi\ in^{2}[/tex]
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
[tex]\frac{64\pi }{360} =\frac{x }{318}\\ \\x=(64\pi)*318/360\\ \\x=56.53\pi\ in^{2}[/tex]
