Question

A football field is surrounded by a track (in light blue) of 4 mile. The track consists of two straight lines and two semi-circles. What length should the straight part of the track be in order to maximize the football field area (the rectangular area inside the track that I've shaded in gray)? Please follow the directions below to do this problem. 2. Label the picture above with at least two variables that represent the relevant lengths. (I suggest that one of them is the radius of the semi-circle in the light blue a) [1 pt] Determine the area of the shaded rectangle in terms of the variables you labeled above. b) [2 pts] Now, use a constraint equation to get your formula in part (b) in terms of one variable

Answer 1

For relative maximum or minimum

To maximize or minimize a function f(x) 1st we need to find its critical points

For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0

Then check second derivative of f(x) i.e. f ''(x) is positive or negative at critical numbers If f ''(x) >0 then that gives minimum value of f(x) and if f ''(x) <0 then that give maximum value of f(x)

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2l 2 t-2 6 22 in Putin A 2 NDw we need masimi2 e disresnhahe A ih espect ho 2 T 2 Valies ol thr coih cal dA diHarennade asan -4m0 T7 Pulh'as =+ I 6 lensh or ne race