What Determines the Maximum Area of an Athletic Field?

[dmonlama]

Homework Statement

An athletic field is to be built in the sahpe of a rectangle x units long capped by semicircular regions of radius r at the two ends. The field is to be bounded by a 400-m racetrack.
a. Express the area of the rectangular portion of the field as a funcion of x alone or r alone (your choice).
b. What values of x and r give the rectangular portion the largest possible area?

The Attempt at a Solution

For a, i expressed the equation in terms of r. I got 40000/pi - pi(r)^2. i just took the overall area and subtract it by the semicircular circles.

 

[HallsofIvy]

What is your question? If you are asking if you did this correctly, I can't say because didn't show how you got that answer. Are you saying that the "overall area" is 4000/pi? How did you get that?

 

[dmonlama]

i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.

 

[Dick]

i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.

No, it's not. If r=0 that gives 40000/pi for the area, which can't be right. Again, show how you reached that conclusion.

 

[HallsofIvy]

What is your question? If you are asking if you did this correctly, I can't say because didn't show how you got that answer. Are you saying that the "overall area" is 4000/pi? How did you get that?

i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.

Now, please answer my original question. HOW did you get "the overall area"?