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
first find the area of the ring (as if the seating area were not there)
the area of the shaded ring without the seating area is\[\text{Area of outer circle}-\text{Area of inner circle}=\pi r_0^2-\pi r_i^2=\pi(r_o^2-r_i^2)\]
im still not fully getting it...
you need the radius if the circles to find their area you are given their diameters what are their radii ?
what is the relationship between a diameter and radius?
the outer circles radious is 55 and the inner circle is 12
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 one|dw:1343338007266:dw|we are calling the outer radius \(r_o\) and the inner radius \(r_i\)
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?\[A_o-A_i=\pi(r_o^2-r_i^2)\]
i got 9046.34
right :) so that is the area of the whole ring now what fraction of a circle is 60 degrees?
1/6
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
i got 1507.5566
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)
so subtract that area from the total area of the ring; which answer is closest?
wait im beginjing to get confused... can you show me?
you almost have the answer you already wrote it, but I'll ask you again: what is the area of the ring?
1507.5566?
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
i got 7538.7834
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 :)
awesome!
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