SOLUTION: If you are given 68 feet of fence what dimensions would give you the maximum area of a rectangular fenced in garden plot?

Algebra ->  Length-and-distance -> SOLUTION: If you are given 68 feet of fence what dimensions would give you the maximum area of a rectangular fenced in garden plot?      Log On


   

Question 378626: If you are given 68 feet of fence what dimensions would give you the maximum area of a rectangular fenced in garden plot?
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The optimum area of a rectangle with a given perimeter is a square, i.e in this case, the side length would be 17 feet and the area would be 289 ft^2.
However, proving this requires some introductory calculus.
If we define a function A%28x%29+=+x%2834-x%29+=+34x+-+x%5E2 then to optimize the value of A(x), we find A'(x) (the derivative of A(x) in terms of x).
By the power rule, A'(x) = 34 - 2x. The function reaches a critical point, or vertex, when A'(x) = 0, or x = 17. It can be verified that values greater or less than 17 will give smaller areas, so x = 17 is the optimum value.