Question

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

Answer 1

Do you know how to find the largest area possible?

Answer 2

Draw a picture

Answer 3

lol

Answer 4

That's fine lol

Answer 5

c=2piR and 200=4r+2x (x being the base of the rectangle) this is how i set it up

Answer 6

i think you have the perimeter wrong

Answer 7

how?

Answer 8

You're not accounting for the circumference of the circle. Your equation only includes the area of the rectangle.

Answer 9

You would need it to be \(2x + c = 200\)

Answer 10

c is the circumference right

Answer 11

Yes, and that's right (\(2 \pi r\)).

Answer 12

okay and then differentiate that equation correct?

Answer 13

You would need to differentiate the area, not the perimeter.

Answer 14

So, can you first determine a formula for the area of your diagram?

Answer 15

pir^2+2rx

Answer 16

Yeah, that looks good.

Answer 17

should i substitute r or x

Answer 18

Whichever one you want

Answer 19

100/3pi = r seem right?