Find Dimension that yield greatest possible Area by completing the square for the rectangular region.

Question

An athletic field with perimeter of 40km consists of a rectangle with a semicircle at each end.

Find Dimensions that yield greatest possible Area by completing the square for the rectangular region. Hint: You need to get r in terms of x.

 

 

Answer 1

Just a guess but usually squares and circles have the maximum area for given perimeter

Let W=L = 2r W=width, L=length and r = semi-circle radius

then Area = 2r(2r) + pi(r)² = (4+pi)r² where 40 = 4r+2rpi = (4+2pi)r

solve for r= 40/(4+2pi) = about 3.9

2r = W = L = 7.8 km