Katie’s Dog Kennel

Question

Katie is building a kennel. A building will form one side. She has 120 feet of fence. Katie wants to enclose the largest area possible. What are the dimensions of the kennel?

Answer 1

So the equation for area of the fenced kennel is A = 120W - 2W²

If you don't yet know about derivatives (usually taught in a calculus class) yet, here's another way of finding the vertex of a parabola:

1. Get the equation in the form y = ax2 + bx + c. (y = -2x² + 120x, for our problem.)
2. Calculate -b / 2a. This is the x-coordinate of the vertex. (-120 / -4, or 30. This is the Width.)
3. To find the y-coordinate of the vertex, plug the value of -b / 2a into the equation for x and solve for y, which gives you y = 60. This is the Length for the maximum enclosed area.

Answer 2

Lets assume the kennel will be a rectangle: width W and length L.

Since one side is a building, the fence is only used on 3 sides, so the total will be 2*W + 1*L

2W + L = 120, or L = 120 - 2W

The area is simply A = L * W

Replace the L with 120 - 2W, and we have:

A = (120 - 2W) * W

A = 120W - 2W²

^{To find a maximum or minimum, take the 1st derivative and set equal to 0:}

^{A' = 120 - 4W = 0}

^{4W = 120}

^{W = 30}

^{from the first equation: L = 60}