Math problem algebra

Question

Stephen 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. The other three sides will be enclosed with wire fencing. If Stephen has 500 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 a function for the area  of the enclosure in terms of . (HINT: First write one equation with w and l and one equation with w and l and A. Solve for l in the first equation and substitute for l in the other).

Answer 1

So this is an optimization problem (a personal favorite of mine), we need 2 equations to solve for what we are looking for, in this case a perimeter and an area, so lets make those:

Perimeter) 2L+2W=500 but one side doesn't need fencing since it is next to a barn, so the actual equation relating to amount of fencing used should be the following: L+2W=500

Area=LW

now lets solve for L in the perimeter equation and plug into the area equation:

L=500-2W ---> Area=W(500-2W)=500W-2W²

Now we have our area equation in terms of w, now lets take the derivative and set it =0 to solve for maximum area:

Area'=500-4W=0, 500=4W, W=125

Now we have w, lets plug it back into l to solve for the length:

L=500-2(125), L=500-250=250

So the Dimensions of the fence should be (LxW) 250x125