Q:

the area of a sector of a circle witha radius measuring 30cm 100 pi cm. what is the measure of the central angle that forms the sector?​

Accepted Solution

A:
Answer:The central angle is [tex]40\Β°[/tex] or [tex]\frac{2}{9}\pi\ radians[/tex]Step-by-step explanation:step 1Find the area of the circleThe area of the circle is equal to[tex]A=\pi r^{2}[/tex]we have[tex]r=30\ cm[/tex]substitute[tex]A=\pi (30)^{2}[/tex][tex]A=900\pi\ cm^{2}[/tex]step 2Find the central angle in degrees for a sector with area [tex]100\pi\ cm^{2}[/tex] Letx----> the measure of the central angle in degreesRemember that the area of the circle subtends a central angle of 360 degreessousing proportion[tex]\frac{900\pi}{360}=\frac{100\pi}{x}\\ \\x=360*100\pi/900\pi \\ \\x=40\Β°[/tex]step 3Find the central angle in radians for a sector with area [tex]100\pi\ cm^{2}[/tex] Letx----> the measure of the central angle in radiansRemember that the area of the circle subtends a central angle of [tex]2\pi[/tex] radianssousing proportion[tex]\frac{900\pi}{2\pi}=\frac{100\pi}{x}\\ \\x=2\pi*100\pi/900\pi \\ \\x=\frac{2}{9}\pi\ radians[/tex]