a rain gutter is being fabricated from a flat sheet of metal so that the cross section of the gutter is a rectangle. The width of the metal is 12 inches.
a. write a formula f(x) that gives the area of the cross section.
b. to hold the greatest amount of rainwater, the cross section should have a maximum area. Find the dimensions that would result in this maximum.
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a) short side of rectangle = x
long side = (12-2x)/2 = 6-x
So f(x) =x(6-x)
thank you...its just x(12-2x)
b) To find maximum you have to see where the derivative gets 0.\[f'(x) = -2x +6\]
it will be 0 for x=3
So that's your answer: short side x=3, long side (12-2x)/2=6-x = 6-3 =3
So it's a square
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what is this (12-2x)/2 ??
The perimeter of rectangle is 12. So if short side length is x, rest this amount twice (since there are two equal sides ) from 12 and the rwhat's left is the sum of the 2 long sides (that's why devide by 2)