Problem 362*This is actually very similar to an A level question but without the added guidance that they usually provide.
A sports association is planning to construct a running track in the shape of a rectangle surmounted by a semicircle. Let the letter
x represents the length of the rectangular section and
r represents the radius of the semicircle.
Determine the length,
x, that
maximises the area enclosed by the track.
EDITThe perimeter of the track must be 600 metres.I completely forgot to put that in. Sorry
600m running track hmm OK different from the 400m standard, nice. Well A=πr2+2rx and P=2x+2πr where P is 600m. There are a number of ways to do this but it would be neatest with lagrange multipliers I think.
A=πr2+2rx
0=2x+2πr−600
A=πr2+2rx+λ(2x+2πr−600)
A=πr2+2rx+2λx+2λπr−600λ
dxdA=drdA=dλdA=0
0=2x+2πr−600
0=2πr+2x+2λπ
0=2r+2λ
r=−λ
0=2πr+2x−2πr
0=2x
This implies the maximum area is enclosed when x = 0.
This makes sense as a circle contains the most area for a given perimeter of any shape.