Find the dimensions of the padlocks so that the fence encloses the largest possible area in the following problem.
A man purchased 2000 m of used wire fencing at an auction. He and his wife want to use the fencing to create three adjacent rectangular padlock.
Let the length of each padlock when the largest area is enclosed be L and the width be W.
The padlocks are adjacent to each other, let the length be shared between them. There is no need to use two fences between each of them, just one is enough. This reduces the total length of fence needed to enclose the 3 padlocks to 6W + 4L.
As the length of the fence available is 2000 m , 6W + 4L = 2000
=> L = (1000 - 3W)/2
The total area enclosed is 3*L*W
=> 3*[(1000 - 3W)/2]*W
=> 1500W - 4.5W^2
To maximize 1500W - 4.5W^2, find the first derivative and solve that for W.
1500 - 9W = 0
=> W = 1500/9
=> W = 166.7 m
L = (1000 - 3W)/2
=> (1000 - 3*(1500/9))/2
=> (1000 - 500)/2
=> L = 250 m
This gives the required dimensions as 250 m and 166.7 m