Question

Suppose you have 54 feet of fencing to enclose a rectangular dog pen. The function a=27x-x^2, where x=width, gives you the area of the dog pen in square feet. What width gives you the maximum area? Round to the nearest tenth as necessary.

Answer 1

For maximum area, we take first derivative and then equalize it to zero
A(x) = 27x- x^2 
A'(x) = 27 - 2x 

Set that equal to zero and solve for x : 
27 - 2x = 0 
27 = 2x ......................... [ added 2x to both sides ] 
13.5 = x ........................ [ divided both sides by 2 ] 

So the area will be
A = 27(13.5) - (13.5)^2 
   = 364.5 - 182.25 
   = 182.3 ft^2