Question

In the semicircular region shaded, DQ = 10''.A circle has its center at point Q.A diameter is horizontal. It starts at point C on the left, goes through point Q and ends at point D on the right.The area above the diameter in the interior of the circle is shaded.(a)Find the exact perimeter of the region in inches. in(b)Find the exact area of the region in square inches. in2

Answer 1

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:

`P=(1)/(2)(2\pi r)+d`

Note that the diameter is twice the radius, that is, d=2r.

Hence, the perimeter becomes:

`P=\pi r+2r`

Substitute r=10 into the formula:

`P=\pi(10)+2(10)=10\pi+20=10(\pi+2)\text{ inches}`

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:

`A=(\pi r^2)/(2)`

Substitute r=10 into the formula:

`A=(\pi(10)^2)/(2)=(100\pi)/(2)=50\pi\text{ in}^2`

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.