SOLUTION: Calculus Optimization Question A rectangular pen is being constructed with 500 feet of fencing. One side of the pen is the wall of a barn and does not require fencing. The oth

Algebra ->  Equations -> SOLUTION: Calculus Optimization Question A rectangular pen is being constructed with 500 feet of fencing. One side of the pen is the wall of a barn and does not require fencing. The oth      Log On


   

Question 1142708: Calculus Optimization Question
A rectangular pen is being constructed with 500 feet of fencing. One side of the pen is the wall of a barn and does not require fencing. The other three sides are fenced. What are the dimensions of the pen that will maximize the area of the pen?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52779) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let x be the dimension of the pen (in feet) perpendicular to the wall. 


Then the side parallel to the wall is  (500-2x)  feet long,


and the area of the pen is


    A(x) = x*(500-2x)  = -2x^2 + 500x  square feet.


They want you find the maximum of the function A(x) using Calculus.


For it, differentiate A(x) over x and equate the derivative to zero:


    A'(x) = -4x + 500 = 0,


which gives you  x = 500%2F4 = 125 feet.


Thus you obtain the


ANSWER.  Under given conditions, the maximum area is achieved for the rectangle 

         with the short side of 125 ft perpendicular to the wall and long side of 500 - 2*125 = 250 ft parallel to the wall.

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I re-wrote/corrected my post after getting a notice from @greenestamps.

@greenestamps, thanks for your notice !



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


For tutor @ikleyn....

Oops! The fence is only on three sides of the pen; the fourth side is the side of a (very long!) barn.