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.

Question

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.

anonymous · Answer

a) short side of rectangle = x
long side = (12-2x)/2 = 6-x
So f(x) =x(6-x)

yrivers36 · Answer

thank you...its just x(12-2x)

anonymous · Answer

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

anonymous · Answer

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)

anonymous · Answer

ok?

yrivers36 · Answer

ok i see