SOLUTION: The Smith's have 160 meters of fencing available to build a rectangular garden. One side of the garden touches a side of the house and doesn't need any bordering. Algebraically fin

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Question 177927: The Smith's have 160 meters of fencing available to build a rectangular garden. One side of the garden touches a side of the house and doesn't need any bordering. Algebraically find the dimensions that will give the maximum 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 house.
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=160
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=160
W=160-2L
2.A=L%2AW
A=L%2A%28160-2L%29
A=-2L%2A2%2B160L
Now we can differentiate with respect to L and set the derivative equal to zero.
dA%2FdL=-4L%2B160=0
L=40
Let's plot the graph of area as a function of length to make sure the area is maximum at this point,
+graph%28+300%2C+300%2C+-20%2C+080%2C+-100%2C+4000%2C+-2x%5E2%2B160x%29+
From eq. 1,
W=160-2L
W=160-2%2840%29
W=80
The area is then,
A=L%2AW=40%2A80=1250
Width of 80 m, length of 40 m, yields a garden of 3200 sq. meters.