Question

The figure below shows the top view of a circular room with a circular stage. The diameter of the stage is 24 feet. The shaded portion represents the seating for the audience around the stage.



[Use π =]

What is the area of the seating portion?

7545.48 ft2

8939.33 ft2

3169.05 ft2

1509.10 ft2

Answer 1

first find the area of the ring (as if the seating area were not there)

Answer 2

the area of the shaded ring without the seating area is\[\text{Area of outer circle}-\text{Area of inner circle}=\pi r_0^2-\pi r_i^2=\pi(r_o^2-r_i^2)\]

Answer 3

im still not fully getting it...

Answer 4

you need the radius if the circles to find their area
you are given their diameters
what are their radii ?

Answer 5

what is the relationship between a diameter and radius?

Answer 6

the outer circles radious is 55 and the inner circle is 12

Answer 7

right
so as I say, to find the area of the shaded part (ignoring the space for the seating area) we subtract the area of the inner circle from the outer onewe are calling the outer radius \(r_o\) and the inner radius \(r_i\)

Answer 8

the area of the outer circle is\[A_o=\pi r_o^2\]the area of the inner circle is\[A_i=\pi r_i^2\]so what is the area of the ring?\[A_o-A_i=\pi(r_o^2-r_i^2)\]

Answer 9

i got 9046.34

Answer 10

right :)
so that is the area of the whole ring
now what fraction of a circle is 60 degrees?

Answer 11

1/6

Answer 12

right, so now we need to take that size chunk out of our ring to find the area of the actual shaded region
so subtract off 1/6 of the area you just found and that's your answer

Answer 13

i got 1507.5566

Answer 14

that is the size of the chunk that we need to remove
(actually I am getting sightly different numbers, but that must be due to the way you are rounding pi which you have not specified, but the answers I'm getting are close enough)

Answer 15

so subtract that area from the total area of the ring; which answer is closest?

Answer 16

wait im beginjing to get confused... can you show me?

Answer 17

you almost have the answer
you already wrote it, but I'll ask you again: what is the area of the ring?

Answer 18

1507.5566?

Answer 19

no, that is the piece that we need to *remove* from the ring
so you need to take the area of the whole ring (which you already found, read above again) and subtract 1507.5566 from that

Answer 20

i got 7538.7834

Answer 21

and I got 7542.440362 by not rounding pi and just punching it into my calculator
either way I think both those answers are close enough to be definitive.
so that's it, we're done :)

Answer 22

awesome!