A rain gutter is made from sheets of aluminum that are 18 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that
will maximize its cross-sectional area and allow the greatest amount of water to flow.
A. 5 inches
B. 4 inches
C. 4.5 inches
D. 5.5 inches
so if we look at the cross section and assume that the amount turned up is the same on both sides
so what we can do is make 1 equation and find the max or vertex
if the sides are legnth x and base is y 2x+y=18
and the cross sectional area is xy
2x+y=18 minus 2x both sides y=18-2x sub for y
x(18-2x)=areamax 18x-2x^2=areamax we have a parabola -2x^2+18x+0=areamax to find the max y valie or area, we find the y value of the vertex but the x value of the vertex is what we are looking for to find the value of x to max the area for ax^2+bx+c=y x value of vertex is -b/2a -18/2(-2)=-18/-4=9/2=4.5 4.5=x value of vertex
