Question

A piece of rectangular sheet metal is 20 inches wide. Its is to be made into a rain gutter by turning up the edges to form parallel sides. let x represent the length of each of the parallel sides. a) Give the restrictions on x b) Determine a function A that gives the area of the cross section of the gutter. C) for what value of x will A be a maximum (and thus maximize the amount of water that the gutter will hold) What is the max area?

Answer 1

Turning up the edges of the sheet metal will result in the 20 inches of metal being divided into x inches of the first side, then the unknown width of the gutter (let's call it "y") and then again x inches for the second side.

 As a picture: x|______|x/y

 As a formula: 20 = x + y + x = 2x + y

Resolving this to y we get: 20 - 2x = y

and switching it around: y = 20 - 2x,

Now for part b) of your question:
The area of the gutter's cross-section is its width (y) multiplied with its height (x).

A(x,y) = x * y
If we use our result from a) to eliminate y then we can see that
A(x) = x * y = x * (20-2x) = 20x-2x^2
So I'd say that the answer to part b) should be: A(x) = 20x - 2x^2