cleThe given region is a semicircle with a radius DQ=10''.
It is required to find the exact perimeter and the exact area.
(a) Recall that the perimeter of a semicircle is the sum of half of the circumference of the circle and its diameter:
[tex]P=\frac{1}{2}(2\pi r)+d[/tex]Note that the diameter is twice the radius, that is, d=2r.
Hence, the perimeter becomes:
[tex]P=\pi r+2r[/tex]Substitute r=10 into the formula:
[tex]P=\pi(10)+2(10)=10\pi+20=10(\pi+2)\text{ inches}[/tex]The exact perimeter of the region is 10(π+2) inches.
(b) The Area of a Semicircle is half the area of a circle given as:
[tex]A=\frac{\pi r^2}{2}[/tex]Substitute r=10 into the formula:
[tex]A=\frac{\pi(10)^2}{2}=\frac{100\pi}{2}=50\pi\text{ in}^2[/tex]The exact area of the region is 50π square inches.
Answers:
The exact perimeter of the region is 10(π+2) inches.
The exact area of the region is 50π square inches.