SOLUTION: A long rectangular sheet of metal 10 in. wide is to be made into a gutter by turning up sides of equal length perpendicular to the sheet. Find the length that must be turned up to

Question 345797: A long rectangular sheet of metal 10 in. wide is to be made into a gutter by turning up sides of equal length perpendicular to the sheet. Find the length that must be turned up to produce a gutter with a maximum cross-sectional area.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A long rectangular sheet of metal 10 in. wide is to be made into a gutter by turning up sides of equal length perpendicular to the sheet. Find the length that must be turned up to produce a gutter with a maximum cross-sectional area.
:
Let x = the depth of the gutter
Let y = the width of the gutter
:
This may illustrate an end view of the gutter: L|_w_|L
then
2L + w = 10 in; the width of the sheets of aluminum
w = (10-2L); use this form for substitution
and
L * w = A; the cross sectional area of the end of the gutter
:
Using the area equation replace w with (10-2L)
:
A = L*(10-2L)
A = -2L^2 + 10L
a quadratic equation, max area occurs at the axis of symmetry, x = -b/(2a)
in this problem x = L; b = 10; a =-2
L =
L =
L = +2.5 inches turned up to create max area
:
:
w = 10 -(2.5)
w = 5 inches is the width
:
Max area would be; 2.5 * 5 = 12.5 sq/inches

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