How do you solve this problem?

Question

A 384 square meter plot of land ts to be enclosed by a fence and divided into two equal parts by another fence parallel to one pair sides. What dimensions of the outer rectangle will minimize the amount of fence used?

Answer 1

Let,

     Length of rectangle = l

     Width of rectangle = w

     Total length of fence = x

So, l*w = 384

     w = 384/l..........(1)

As the area must be divided by a fence parallel to the 2 shortest sides, let the shortest side be w and the longest be l

x = 3l + 2w

   = 3l + 768/l [Plug in w from (1)]

Take the derivative,

x' = 3-768/l²

Taking x' = 0,

3-768/l² = 0

3 = 768/l² 
3*l² = 768
l² = 256 

l = 16

Plug l into equation (1) and solve for w,

w = 384/16

   = 24

Hence, an outer rectangle of 16m*24m will minimize the amount of fence used