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

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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.

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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.