Question 425090
You are designing a rain gutter made from a piece of sheet metal 1 foot by 5 feet.
 The gutter will be formed by turning up two sides.
 You want the gutter to have the greatest volume possible.
;
a.How much should you turn up? 
Change the width of the sheet metal to 12 inches
:
The end of the gutter sort of looks like this x|__|x
x = the height of the gutter, (the amt turned up)
then
12 - 2x = the width of the gutter
:
Greatest volume with be determined by the greatest cross section area
A = x(12-2x)
A = -2x^2 + 12x
Find the axis of symmetry to find the height with the greatest area
x = -b/(2a); in this equation a=-2, b=12
x = -12/(2*-2)
x = 3 inches is the height for the greatest cross sectional area
then
12 - 2(3) = 6 inches is the width of the gutter
3 * 6 = 18 sq inches is the cross sectional area
:
b. What is the maximum volume?
18 *5(12) = 1080 cu/inches
or
{{{1080/12^3}}} = .625 cu ft