precalc
posted by steph on .
construction of a rain gutter: a piece of rectangular sheet metal is 20 inches wide. it is to be made into a rain gutter by turning the edges to form parallel sides. let x represent the length of each of the parallel.
a) give the restrictions on x.
b) determine a function of A that gives the area of a cross section of the gutter.
c) for what values of x will A be a maximum (and thus maximize the amount of the water that the gutter will hold)?
d) what is this maximum area?
e) for what values of x will the area of a cross section be less than 40 square inches?

Look at a crosssection of the gutter.
let the height be x
then the base width = 202x , assuming the gutter is open at the top
a) in 202x , 0 < x < 10
b) Area = x(202x)
c) Area = 20x  2x^2
= 2(x^2  10x + 25  25 )
= 2( (x5)^2  25)
= 2(x5)^2 + 50
d) maximum area is 50 when x = 5
e) x(202x) < 40
2x^2 + 20x < 40
x^2  10x + 20 > 0
consider x^2  10x + 20 = 0
x = (10 ± √20)/2
= 5 ± √5
so the area is less than 40 for
5√5 < x < 5+√5 
thank you! that helped a lot

how did you figure c?