SOLUTION: one hundred feet of fencing is to be used to enclose a rectangular garden that abuts to a barn. No fencing is needed along the barn. What is the largest possible area that can be

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: one hundred feet of fencing is to be used to enclose a rectangular garden that abuts to a barn. No fencing is needed along the barn. What is the largest possible area that can be      Log On


   

Question 332820: one hundred feet of fencing is to be used to enclose a rectangular garden that abuts to a barn. No fencing is needed along the barn. What is the largest possible area that can be enclosed?
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
Perimeter of the rectangular garden = 2*W+L (one side is covered by the barn)
The area of the garden = L*W
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so we know we have a total of 100 feet of fencing
therefore, 100+=2%2AW%2BL
solve this for L: L=100-2%2AW
substitute the L into the area formula : A=L*W
A=L%2AW=%28100-2%2AW%29%2AW
A=+100%2AW+-2%2AW%5E2
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To find the minimum or maximum area, take the derivative and set to 0
why? because a min or max will occur when the slope is 0, flat part of the function.
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dA/dW = 100 -4*w =0, solve for W
W=100/4=25 is this a min or max? if second derivative at w=25 is negative then its a maximum, why?
because the second derivative relates to the curvature of the function, so if its negative its curving downward
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d^2A/dW^2= -4 thus function is curving downward, W=25 is a maximum
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now W=25 means that L=100-2*W=50
therefore
max Area=L*W=25*50=1250 ft^2