Length/Width math problem

Question

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and the width of the plot that will maximize the area. What is the largest area that can be enclosed?
I've been having a lot of trouble with setting this equation up since there are so few numbers provided.

Answer 1

Note that this is still a rectangle, and the length of the fencing is still 600 ft.

                                         river

Visualize it like this:~~~~~~~~~~~~~~~~~

                                 |                       |

                              x |                       | x

                                 |_____________|

                                            y

 

So the sum of the sides NOT along the river   x + x + y = 600,  and the area equals xy.

 

This makes the two equations:    2x + y = 600,   and A = xy.

 

To find the largest area, we need to find A as a function of x or y.  I suggest solving the first equation for y and replacing that in the second equation.

 

                                y = 600 - 2x.    and A(x) = x(600-2x)

 

          We now need to maximize A(x) = 600x - 2x².

 

Remember, if x = -b/(2a), we find the x value of the vertex, the y value can be found by substitution.

 

So, since a = -2, and b = 600, x = -600/(-4) = 150 ft.  If x = 150, y = 600 -  2(150) = 300.

 

So, the dimensions are 150 x 300 and the maximum area = 300(150) = 45,000 ft²

 

I hope this helps.  By the way, there are many variations of this, and they are all similar.  For example, you might want to make several pens with two lengths parallel and have three parallel withs inside.