Please Help I am lost

Question

A child-care center has 200 feet of fencing to enclose two adjacent rectangular safe play areas. The length is 2x, and the width is y. Another fence along the width bisects the area, separating it into two areas. Use the following methods to determine the dimensions that will produce a maximum enclosed area.

a) Write the total area A of the play areas as a function of x.

b) Use the table feature of a graphing utility to create a table showing possible values of x and the corresponding total area A of the play areas. Use the table to estimate the dimensions that will produce the maximum enclosed area.

c) Use the graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area.

d) Write the area function in standard form to find algebraically the dimensions that will produce the maximum enclosed area.

Answer 1

So the first thing we will want to do with a problem like this is to draw a picture! So we are building a fence to make two separate play areas and the middle part of the fence bisects the area, which means it splits it into two equal halves, so the picture would look like this:

_____________2x______________

|................... |...................... |

|................... |...................... |

|................... |...................... | y

|................... |.......................|

|____________|_____________ |

(Ignore the dotted parts. The chat box wouldn't let me just have a bunch of blank space, so I had to fill it in with something)

a) We want to write the area as a function of x. So how do we find the area of a rectangle? We multiply the length times the width, which according to our picture would give us A = 2x * y. However, the problem is that we have a "y" in our equation, but we want it to only be in terms of x! So we have to think of a way to rewrite "y" in terms of "x".

So the key here is to think about how much fencing we have in total, which is 200 feet. Now, if I use 2x feet of fencing on the top and 2x feet of fencing on the bottom, how much would be left for the vertical portions of the fence? Take that answer and divide it by 3 because we have 3 vertical sections of fence, and then boom, you have your "y" in terms of "x". Once you have "y" in terms of "x", then you can substitute it into our original area equation, A = 2x * y. Then, you will have the area function in terms of x.

b) and c) Once you have your area equation in terms of "x", just plug it into your calculator and do exactly what these parts say to do.

d) Your answer in part a) for the area will be a quadratic equation. Remember that the standard form of a quadratic equation is Ax² + Bx + C = 0. Once you put it into that form, then you can use x = -B/(2A) to give you the x-value of the maximum point on your graph. Once you know "x", then you can plug it in to find the dimensions of the rectangle.

Those are some good hints to get you started. Good luck!