# You have 400 feet of fencing to construct a rectangular pen for cattle. What are the dimensions...

## Question:

You have 400 feet of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximizes the area?

## Finding Maxima And Minima Of A Function:

If we have a continuous function {eq}\displaystyle f(x) {/eq} then its absolute extrema are found by evaluating the function value at its critical points which are obtained by equating its first derivative w.r.t. x to 0 i.e. {eq}\displaystyle f'(x)=0 {/eq}. The values of x obtained, say {eq}\displaystyle x_1,x_2,x_3,\dots {/eq} will give the critical points. At each of the critical points, there could a potential maxima or minima. To verify this we calculate the second derivative of the function at these critical points. If {eq}\displaystyle f' '(x)\lt 0 {/eq} at any critical point then the function will be a maxima at that point, but if {eq}\displaystyle f' '(x)\gt 0 {/eq} then the function will have a minima that point.

• Let the pen have a width of {eq}\displaystyle x {/eq} and a height of {eq}\displaystyle y {/eq}.

• So the total perimeter of the pen will be given by, {eq}\displaystyle x+y+x+y=2(x+y) {/eq}.

• But the pen will be made from a {eq}\displaystyle 400 {/eq} ft long fencing.

• Hence we have,

$$\displaystyle 2(x+y)=400\quad \quad \Rightarrow \quad \quad x+y=200\quad \quad \Rightarrow \quad \quad \boxed{y=200-x}$$.

• Now the area bounded between the rectangular pen will be given by, {eq}\displaystyle A=xy=x(200-x)=200x-x^2 {/eq}.

• To maximize it we need to differentiate and equate to zero to find the critical points.

\displaystyle \begin{align} &\frac{d}{dx}[A]=200-2x\\ \Rightarrow &200-2x=0\\ \Rightarrow &x=100 \end{align}

• Further let us confirm that this value of x will indeed give a maxima. for this we find the value of second derivative at {eq}\displaystyle x=100 {/eq}

$$\displaystyle \frac{d^2}{dx^2}[A]=0-2=-2 \lt 0$$

• So the second derivative will be -ve for any x value, hence {eq}\displaystyle x=100 {/eq} will indeed be a maxima.

• Thus we get {eq}\displaystyle y=200-(100)=100 {/eq}.

• So the size of the pen must be {eq}\displaystyle 100\times 100 {/eq} to get maxima area.