Rain Gutters

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Chapter 3, Section 3.4

[Eigenvalue]

Let the cross-sectional area be represented as A(x)
A(x)=x(12-2x)=12x-2x²

a) The maximum cross-sectional area would be the maximum value for A(x), or the vertex
x=[tex]\frac{-b}{2a}[/tex]=[tex]\frac{-12}{2(-2)}[/tex]=3
A(3)=12(3)-2(3)²=18
A depth of 3 inches will allow for the maximum cross-sectional area.

b) A(x)[tex]\ge[/tex]16
-2x²+12x[tex]\ge[/tex]16
-2x²+12x-16[tex]\ge[/tex]0
x²-6x+8[tex]\le[/tex]0 (note: dividing by a negative value necessitates a change in the inequality symbol)
(x-4)(x-2)[tex]\le[/tex]0
2[tex]\le[/tex]x[tex]\le[/tex]4
Depths of 2,3, or 4 inches would allow for at least 16 square inches of water flow.