SOLUTION: A roll of aluminum with a width of 32 cm id to be bent into rain gutters by folding up two sides at 90 degree angles. A rain gutter's greatest capacity, or volume, is determined by

Question 854548: A roll of aluminum with a width of 32 cm id to be bent into rain gutters by folding up two sides at 90 degree angles. A rain gutter's greatest capacity, or volume, is determined by the gutter's greatest cross-sectional area.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A roll of aluminum with a width of 32 cm id to be bent into rain gutters by folding up two sides at 90 degree angles.
A rain gutter's greatest capacity, or volume, is determined by the gutter's greatest cross-sectional area.
:
Let s = the length of the 2 sides
Let w = the width of the bottom of the gutter
then
2s + w = 32 cm
w =(32-2s), use this form for substitution
Area
A = s*w
replace w with (32-2s)
A = s(32-2s)
a quadratic equation
A = -2s^2 + 32s
Max area occurs on the axis of symmetry, which is x = -b/(2a)
In this problem x=s; a=-2, b=32
s =
s = +8 cm is the length of the side
then
w = 32 - 2(8)
w = 16 cm is the width of the bottom
:
Actual cross-sectional area: 8 * 16 = 128 sq/cm, max area.
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