Question

A sheet of metal that is 30 cm wide and 6 m long is to be used to make a rectangular eavestrough by bending the sheet along the dotted lines. What value of x maximizes the capacity of the eavestrough?

Answer 1

where are the dotted lines on the sheet?

Answer 2

the capacity depends on the area that you get when cutting along the eavestrough and looking the direction the water travels.

Answer 3

we can model the dependence of area, sides x and floor side
area f(x) = x * (30-2x)

where
x: one type of sides in the open rectangle formed with the metal
30-2x: the other type of side in rectangle (floor and open ceiling)

Answer 4

whenever we increase x, we decrease the floor. if we did not fold
any x=0, then the floor still has the max of 30 cm.

if we want a canal, we need to take away from the floor metal two equal parts to make "walls" left and right.
therefore, whenever we increase the wall height by x, we decrease floor width by 2x because it has to donate to left AND right: 30-2x

the metal is only 30 cm in total width, so it's a zero-sum game: whenever we increase x, we decrease the floor. that's why the 30 constant is in the eq.