Question

A farmer wishes to enclose a rectangular field of area 450 ft using an existing wall as one of the sides. The cost of the fence for the other 3 sides is $3 per foot. How do you find the dimensions that minimize the cost of the fence?

Answer 1

`a=15, b=30`

Explanation:

The cost function is:

`y(a,b)=3(2a+b)`

since farmer is using the wall of length `b`.
The area of the field is:

`450=a*b => b=450/a`

`y(a)=3(2a+450/a)=6a+1350/a`

`y'(a)=6-1350/a^2=(6a^2-1350)/a^2`

`y'(a)=0 <=> 6a^2-1350=0 <=> 6a^2=1350`

`a^2=1350/6=225 => a=15` since `a>0` because it's length.

`AAa in (0,15): y'<0` and `y` is decreasing
`AAa in (15,oo): y'>0` and `y` is increasing

So, for `a=15` function `y` has a minimum value `y_min=y(15)=180`

The dimensions of the field:

`b=450/a=450/15=30`

`a=15, b=30`