Question

A rectangular garden laid out along your neighbour's lot contains 432m^2. It is to be fenced on all sides. If the neighbour pays for half the shared fence, what should be the dimensions of the garden so that your cost is a minimum?

Answer 1

`24` m along the boundary with the neighbor's property and `18` m onto my property.

Explanation:

Let `l` be the length of the fence along the property line and `w` the depth onto my property.

The amount of fence that I must pay for is `1/2l+w+l+w = 2w+3/2l`

The area is to be `432` m`""^2` making `wl = 432`, so `w = 432/l`

The fence I must pay for totals (in meters):

`2(432/l)+3/2l = 864/l + 3/2l`

To minimize my cost, I will minimize the amount of fence I must pay for. So I want to find `l` and `w` to make

`f(l) = 864/l + 3/2l` as small as possible.

`f'(l) = -864/l^2+3/2 = (3l^2-1728)/(2l^2)= (3(l^2-576))/(2l^2)`

`f'(l) = 0` at `l = +-sqrt(576) = +-24`.

Clearly `l >= 0`, so we need only consider `l = 24`.

By either the first or second derivative test `f(24)` is a relative minimum and by "the only critical number in town test", we can change "relative" to "absolute".