Question 39262: Bob has 3000 feet of fencing available to enclose a rectangular field.
a) express the area A of rectangle as a function of x where x is the length of rectangle.
b) For what value of x is the area largest?
c) What is the maximum area?
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! a) Length is L = 3000
If one side of the rectangle is x and the other side is y, then perimeter is
P = 2x + 2y
But perimeter P = length of fencing = L
.: P = L = 3000
2x + 2y = 3000
x + y = 1500
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Area of rectangle, A = x*y
A = xy
substitute for y = 1500 - x,
A = x(1500 - x)
A = 1500x - x²
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b) To get the largest area, find the turning points of A(x) = 1500x - x² to get a maximum.
dA/dx = 1500 - 2x
when dA/dx = 0, then A is a maximum or minimum
0 = 1500 - 2x
2x = 1500
x = 750
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There is a turning point at x = 750.
d²A/dx² = -2
Since d²A/dx² is negative (at the turning point x = 750) then the turning point is a maximum.
So max area is gotten when x = 750.
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c) A = x(1500 - x)
A = 750(1500 - 750)
A = 750²
A = 562,500
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