1. ## Differentiation Applications

So I am having some trouble with a certian type of question that has arisen in some of the past work I have done leading me to believe that it could appear in my upcoming examination. So on with the question:

A farmer plans to use a river as a one boundary of a rectangular paddock. If the farmer has 480m of fencing to be used to fence the remaining 3 sides, what dimensions should the paddock be to ensure maximum area.

Help is greatly appriciated.

2. Originally Posted by holmesb
...

A farmer plans to use a river as a one boundary of a rectangular paddock. If the farmer has 480m of fencing to be used to fence the remaining 3 sides, what dimensions should the paddock be to ensure maximum area.
Hello,

let w be the width of the rectangle and
l be the length of the rectangle.

Then the area is calculated by:

$a = w \cdot l$ [1]

the fence f consists of $f = 2\cdot w + l$. Since f = 480 you get:

$l = 480 - 2 \cdot w$ [2]

Plug in the term for l into the equation [1]:

$a(w)=w \cdot (480-2 \cdot w) = -2\cdot w^2 + 480w$

This is a parabola opening downward, the maximum value is at it's vertex:

$a'(w) = -4\cdot w + 480$ and $a'(w) = 0$

You'll get: w = 120 m and l = 240 m. The maximum area is 28800 m²