Question

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?

Answer 1

Look at a cross-section of the gutter.
let the height be x
then the base width = 20-2x , assuming the gutter is open at the top

a) in 20-2x , 0 < x < 10
b) Area = x(20-2x)
c) Area = 20x - 2x^2
= -2(x^2 - 10x <b>+ 25 - 25 </b>)
= -2( (x-5)^2 - 25)
= -2(x-5)^2 + 50
d) maximum area is 50 when x = 5

e) x(20-2x) < 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

Answer 2

thank you! that helped a lot

Answer 3

how did you figure c?