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 paral
lel 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?
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
