Question

A simple rain gutter is constructed using a sheet of aluminum that is 40cm wide. The edges are turned up to form right angles. Determine the depth of the gutter that will maximize the cross-sectional area (allowing the greatest amount of water to flow).

I don't know where and how to start.

please help and thank you

Answer 1

make a sketch.
Assuming that the gutter is rectangular and the sides must have the same height, look at the cross section of the gutter.
let the base by y and each of the sides by x

we know that
2x + y = 40
y = 40-2x

the area A of the cross-section will be
A = xy
= x(40-2x) = 40x - 2x^2
Assuming this is a typical Calculus question ...

dA/dx = 40 - 4x = 0 for a max of A
40-4x = 0
x = 10
the the depth will be 10 cm

if you don't know Calculus , you will have to complete the square on
A = 40x - 2x^2

Answer 2

I don't know calculus

This is what I got after completing the square:
A=-2(x-10)^2+200

How do I know what is the depth when I complete the square?

thank you