Problema Solution
A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is to be 1600 meters. What should the dimensions be so that the area of the rectangle is maximized?
Answer provided by our tutors
let
l = the length of the rectangle
w = the width of the rectangle
Perimeter of the track
P = 2pi(w/2) + 2l
2pi(w/2) + 2l = 1500
w = (1500 - 2l)/pi
Area of Rectangle A = l*w
A = ((1500 - 2l)/pi)*l
A = (1500l - 2l^2)/pi
A = - (2/pi)l^2 + (1500/pi)l
the vertex point, being a maximum or minimum of the function, can be obtained by
l max = -b/2a
l max = (-1500/pi)/(2(-2/pi))
l max = 1500/4
l max = 375 m
w max = (1500 - 2*l max)/pi
w max = (1500 - 2*375)/pi
w max = 750/pi
w max = 238.85 m
the dimensions should be 375 m and 238.85 m.