SOLUTION: Marshall has 36 feet of fencing to use to build a pen. He plans to use the barn for on side of the pen. What is the largest area he can create for the pen using the 36 feet of fe

Question 33204: Marshall has 36 feet of fencing to use to build a pen. He plans to use the barn for on side of the pen. What is the largest area he can create for the pen using the 36 feet of fencing and the side of the barn?
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
Draw a straight line to denote the wall. Draw three more straight lines to create a rectangle.

Let 2 lengths of pen coming off the wall both be x.
The third length of te pen is therefore 36-2x in length.

Area, A = x(36-2x)

Now there are 2 methods to find the maximum value.

1. Find the roots of this equation (where the curve crosses the x-axis). The maximum point on the curve will then lie equidistant from these two, since ALL parabolas are symmetric in shape.

2. Use differentiation to find the maximum value directly.

--------------------------------------------------------------------------
1.

so x=0 or 36-2x = 0
x=0 or 2x=36
x=0 or x=18

You can see this if i plot the graph:

By knowing that a quadratic is symmetrical about the turning point, then roots are at x=0 and x=18, so the turning point, the maximum here, is at x=9.

So dimensions of the pen are 9x18. This makes the max area 162.

2.

differentiate to find the turning point directly:

4x = 36
--> x = 9

So, the maximum is at x=9. Area is as before

jon.

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