Kyle wants to build a rectangular enclosure for his animals. One side of the pen will be against the barn, so he needs no fence on that side.

Question

The other three sides will be enclosed with wire fencing. If Kyle has 450 feet of fencing, you can find the dimensions that maximize the area of the enclosure.

a) Let  be the width of the enclosure (perpendicular to the barn) and let  be the length of the enclosure (parallel to the barn). Write an function for the area  of the enclosure in terms of . (HINT first write two equations with  and  and . Solve for  in one equation and substitute for  in the other).

A(w)=_____ 

b) What width  would maximize the area? 

w= _____ ft

c) What is the maximum area? 

A=_____ square feet

Answer 1

Part A

A = L*W

Perimeter P = L + barn + 2W

barn = 0ft (no fence)

L + 2W = 450ft

L = 450 - 2W

A = (450 - 2W)*W

= 450W - 2W²

A = -2W² + 450W

Part B

Since our equation for area in terms of width takes the form ax² + bx + c (where C = 0), we know we have a parabola. Since the coefficient on the squared term is negative (a = -2), we know we have a parabola that opens downward. This means that we will have a maximum point at the vertex of the parabola, which is our maximum area.

We will say the vertex is at point (h, k), from the function y = (x - h)² + k, in which k is our maximum area and h is our maximizing number: in this case, the value for width that gives us our maximum area.

We know h = -b / 2a. (I can go through the steps to get here if necessary, but it is good to have this equation memorized).

Therefore: h = -450 / 2(-2) = -450/-4

= maximum width = 112.5 ft

Part C

We can use an equation to solve for k (or check our graphing calculator if permitted to use one), but I find it easier to just plug the maximizing value for width (from part B) into our equation for area (from part A).

A = -2(112.5)² + 450(112.5)

maximum area A = 25,312.5 ft²

We can double check our work by solving for maximum length and checking that it matches for length with the given value of perimeter from our solved value for maximizing width, and for length with our solved value for maximum area.

perimeter P = 2W + L

450ft = 2(112.5ft) + L

L = 225ft

Now check maximum area with this length and our solved width:

A = L*W

25,312.5ft² = (225ft)*(112.5ft)

25,312.5ft² = (225ft)*(112.5ft)