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...

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