## yrivers36 3 years ago 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.

1. myko

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

2. yrivers36

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

3. myko

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

4. myko

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)

5. myko

ok?

6. yrivers36

ok i see