Question

A farmer has 25 yards of fencing to make a pig pen. He is going to use the side of the barn as one of the sides of the fence, so he only needs to fence 3 sides. What should be the dimensions of the fence in order to maximize the area?

Answer 1

\[\sf \text{Area of the fence} = l \cdot w = x(25-2x)\]

Answer 2

Let w, L and A be the width, length and area of the fenced in pig pen.
Then:
{2 w + L = 25, A = w L}
L = 25 - 2w
A = w( 25 - 2w)
Set the derivative of A to zero and solve for w.
In the end:\[\left\{w=\frac{25}{4},L=\frac{25}{2}\right\} \]

Answer 3

\[\sf A = -2x^2 -25x\]It's asking you to find the maximum area so you're trying to find the height (y) of your parabola at whatever x may be. To find the x-value of the vertex of the parabola, you use:\[\sf x= -\frac{b}{2a}\]

Answer 4

\[\sf b= -25 ~,~ a=-2\]\[\sf x = \frac{-(-25)}{2(-2)} = \frac{25}{4}\]Use this x value to find the maximum height, \(\sf f\left(\frac{25}{4}\right)\)

Answer 5

How big is the barn?

Answer 6

\[\sf A\left(\frac{25}{4}\right) = -2\left(\frac{25}{4}\right)^2 -25\left(\frac{25}{4}\right)\]