Posted by **steph** on Thursday, November 10, 2011 at 8:28pm.

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?

- pre-calc -
**Reiny**, Thursday, November 10, 2011 at 9:38pm
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 **+ 25 - 25 **)

= -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

- pre-calc -
**steph**, Thursday, November 10, 2011 at 9:47pm
thank you! that helped a lot

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