4096: Solve this problem algebraically

Question

Solve this problem algebraically, using your knowledge of derivatives. Do not solve by graphing.
If 200 feet of fencing is used to enclose a rectangular plot of land that borders a river, what is the maximum area that can be enclosed? Answer to the nearest square foot without commas. For example, if the answer is 1,000, write 1000.

Answer 1

Let x = the length of a side that is perpendicular to the river (there will be two of them)

That means the side parallel to the river will have a length represented by 200 - 2x.

 

Now we can write the area as a function of x.

 

A = x(200 - 2x) = 200x - 2x²

 

So A' = 200 - 4x

Setting that equal to zero results in x = 50. So the parallel side is 100.

 

Therefore the area is 1500 sq ft.

 

For completeness we should show that this is indeed a maximum using either the first or 2nd derivative test.