Question

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

Answer 1

The depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow is 4.5 inches.

Step-by-step explanation:

To determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow, we need to consider the dimensions of the gutter and the factors that affect its cross-sectional area.

The dimensions of the gutter are given as 18 inches wide. Let's assume the depth of the gutter is x inches.

The cross-sectional area of the gutter can be calculated as:

A = (18 - 2x) * x

To find the depth that maximizes the area, we can take the derivative of A with respect to x and find its critical points. By setting the derivative equal to 0 and solving for x, we can find the depth that maximizes the area.

Taking the derivative of A with respect to x:

dA/dx = 18 - 4x

Setting the derivative equal to 0:

18 - 4x = 0

4x = 18

x = 4.5

Therefore, the depth of the gutter that will maximize its cross-sectional area is 4.5 inches, so the correct answer is C. 4.5 inches.

Answer 2

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