answer a, b and c

Question

You will build a rectangular sheep pen next to a river. There is no need to build a fence along the river, so you only need to build three sides. You have a total of  feet of fence to use. Find the dimensions of the pen such that you can enclose the maximum area.

Let the width be  ft, and the length be  ft. Since there are only three sides of the fence, we have:

If the pen’s width is  feet, then its length is  feet. Now we can build a function for the area of the pen:

In this formula,  represents the pen’s width in feet, and  represents the pen’s area in square feet.

Answer the following questions:

A.) To maximize the area of the pen, the length of the pen (parallel to the river) should be?

B.) To maximize the area of the pen, the width of the pen (away from the river) should be?

C.) The maximum area of the pen is?

Answer 1

Recall the formulas for perimeter and area of a rectangle. Now adjust the perimeter formula to account for the missing side. Keeping with the wording of the question it is one of the lengths that is missing. So two widths and one length must add up to the given amount of fencing, lets call the amount of fencing P.

Thus P = 2W+L.

We can use this information to solve for one of the missing variables in terms of the other. L=P-2W.

And since area of a rectangle is given by A=L*W, we can construct a formula for the area of the pen.

A=L*W, but we know L=P-2W, subbing this in we get

A=(P-2W)W

Or

A=P*W - 2W² (A quadratic!)

A quick application of the vertex formula (-b/(2a), f(-b/(2a))) will give us the W that maximizes the area. (-b/(2a)) and the maximized Area ( f(-b/(2a)). To get the missing value of L plug the W value into the L=P-2W formula.