# Lacinda has 120 ft of fencing to make a rectangular kennel for her dogs. The house is to be used...

## Question:

Lacinda has 120 ft of fencing to make a rectangular kennel for her dogs. The house is to be used as one side of the kennel. What length should the kennel be to maximize its area?

## Maxima/Minima of a Function:

Given a function f(x), if it attains a maximum or minimum at a point a, then {eq}\frac{df}{dx} |_{x=a} =0 {/eq}

To know whether the point gives a maximum or a minimum, we can apply the second derivative test at that point. If the second derivative at that point is negative, then the point corresponds to a maximum. If the second derivative at the point is positive, then it corresponds to a minimum.

Let the dimensions of the rectangle be x, y

Given {eq}2(x+y) = 120 \\ y = 60-x {/eq}

Required to maximise area :

{eq}A = xy = x(60-x) = 60x - x^2 \\ \frac{dA}{dx} = 0 \\ 60 - 2x = 0 \\ x = 30 ft \\ y = 30 ft {/eq}

In order to maximise the area, length of the kennel has to be 30 ft 