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optimization problem: a physical fitness room consists of a rectangular region with a semicircle on each end. if the perimeter of the room is to be a 200 meter running track, find the dimensions that will make the are of the rectangular region as large as possible

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Do you know how to find the largest area possible?
Draw a picture

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Other answers:

That's fine lol
c=2piR and 200=4r+2x (x being the base of the rectangle) this is how i set it up
i think you have the perimeter wrong
You're not accounting for the circumference of the circle. Your equation only includes the area of the rectangle.
You would need it to be \(2x + c = 200\)
c is the circumference right
Yes, and that's right (\(2 \pi r\)).
okay and then differentiate that equation correct?
You would need to differentiate the area, not the perimeter.
So, can you first determine a formula for the area of your diagram?
Yeah, that looks good.
should i substitute r or x
Whichever one you want
100/3pi = r seem right?

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