1. The problem statement, all variables and given/known data Maximize the area (in feet) of the rectangular field inside of a mile long racetrack. 2. Relevant equations Circumference of a circle = 2πr P= 2x + 2y 3. The attempt at a solution Area of the semicircles = πr^2 Area of the rectangle = 2rh A(r) = πr^2 +2rh P= 2πr + 2h + 4r 5280ft = 2πr + 2h + 4r h= 2640 - πr - 2r A(r) = πr^2 + 2r(2640 - πr - 2r) A(r) = πr^2 + 5280r - 2πr^2 - 4r^2 A(r) = 5280r - πr^2 - 4r^2 A'(r) = 5280 - 2πr -8r 0 = 5280 - 2πr -8r 2πr + 8r = 5280 r(2π + 8) = 5280 r = 5280/(2π + 8) ft r ≈ 369.67 ft A"(r) = -2π - 8 A"(r) < 0 r is a maximum h = 2640 - (5280/(2π + 8))π - 2(5280/(2π + 8)) h = 10560/(2π + 8) ft h ≈ 739.33 ft A(r) = πr^2 +2rh A(r) = π(5280/(2π + 8) ft)^2 + 2((5280/(2π + 8) ft)(10560/(2π + 8) ft)) A(r) ≈ 975,917 ft^2 I believe this answer is incorrect because for the perimeter, I included the two diameters of the semicircles (the width of the imaginary rectangle). So I ended up using part of the given distance to make these sides and my perimeter formula does not represent the actual perimeter of the track. I tried to redo this problem just using: P= 2πr + 2h however this formula produces an r value that makes the h value equal zero. Obviously I am missing a small detail and I have no idea what it is!