Arain 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

aswer is 4.5in

C is answer