how do i do this polar coordintate problem?

Question

polar coordinates can be used to model a concert amphiteheater. suppose the performer is placed at the pole and faces the direction o the polar ais. the seats have been built to occupy the region with -180(radians) divided by 3 < theta <180 divded by 3 and .25 sketch this region in the polar plane.
how many seats are there if each person has 6 square feet of space?
pleaseeee help thank you so much!

Answer 1

I bet you mean 180°, not radians. Then it would go from -180°/3 = -60° to 180°/3 = 60°, for a total of 120°.

The picture would look a bit like a rainbow, with angled ends. Or a partially eaten pizza slice, where the person started eating at the point but hasn't gotten all the way back to the crust.

To find the area, you can find the area of a circle with radius 300 feet (90000π ft²), subtract the area of a circle with radius 25 ft(625π ft²), and multiply by 120°/360°, or 1/3:

1/3(90000 - 625)π = 89375/3 π = 93593 ft²

So you could accommodate 93593 ft² x (1 person/6 ft²) = 15,598 people (call it 15,600).