SOLUTION: A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100ft of fence? W

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Question 176180: A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100ft of fence? What should the dimensions of the garden be to give this area?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
One side of the rectangle is taken up by the barn.
That leaves 3 sides taken up by fencing.
Let's call that side W for width and the other two sides will be L to use up the fencing.
1.2L%2BW=100
The area of the rectangle will be
2.A=L%2AW
Use eq. 1 and make the area a function of only one variable.
1.2L%2BW=100
W=100-2L
2.A=L%2AW
A=L%2A%28100-2L%29
A=-2L%2A2%2B100L
Now we can differentiate with respect to L and set the derivative equal to zero.
dA%2FdL=-4L%2B100=0
L=25
Let's plot the graph to make sure the area is maximum at this point,
+graph%28+300%2C+300%2C+-10%2C+050%2C+-100%2C+1300%2C+-2x%5E2%2B100x%29+
From eq. 1,
W=100-2L
W=100-2%2825%29
W=50
The area is then,
A=L%2AW=25%2A50=1250
Width of 50 ft, length of 25 ft, yields a garden of 1250 sq. feet.