A 10-foot section of gutter is made from a 12-inch-wide strip of sheet metal by folding up 4-inch strips on each side so that they make the same angle with the bottom of the gutter. Determine the depth of the gutter that has the greatest carrying capacity.

If the angle is x, then you have a trapezoid with lower base 4", and upper base 4+2*4cosx, and height 4sinx.

So, the area of the cross-section is

a = (4 + (4+8cosx))/2 * (4sinx)
= 16 sinx (1+cosx)

da/dx = 16(2cos^2(x)+cos(x)-1)
= 16(2cosx-1)(cosx+1)

da/dx=0 when cosx = 1/2, or x = 60°

Check to be sure it is a max, not a min.

To determine the depth of the gutter that has the greatest carrying capacity, we need to understand that the carrying capacity of a gutter depends on its cross-sectional area.

Let's break down the problem step by step.

Step 1: Calculate the length of the folded strip
The 10-foot section of gutter is made from a 12-inch-wide strip of sheet metal. Since the width of the strip is given in inches, let's convert 10 feet to inches:
10 feet = 10 * 12 inches/foot = 120 inches.

Step 2: Calculate the remaining unfolded width
Since 4-inch strips are folded up on each side, we need to subtract that from the width of the strip:
12 inches - (4 inches + 4 inches) = 12 inches - 8 inches = 4 inches.

Step 3: Calculate the depth of the gutter
The depth of the gutter is the height of the folded portion. We need to determine the depth that maximizes the cross-sectional area of the gutter.

Since the folded strips make the same angle with the bottom of the gutter, we can assume that the bottom width of the gutter is the same as the unfolded width (4 inches). Considering the symmetry, the depth will be the same on both sides, making the cross-sectional shape of the gutter an isosceles triangle.

Now, we can use the formula for the area of an isosceles triangle, which is A = (1/4) * b * h^2, where A is the area, b is the base (unfolded width), and h is the height (depth).

We need to maximize A, so we can rewrite the equation in terms of h:
A = (1/4) * b * h^2
A = (1/4) * 4 * h^2
A = h^2

Step 4: Calculate the maximum height (depth) of the gutter
To find the maximum depth of the gutter, we have to maximize the area. In a triangle, the area is maximized when the height is at its maximum.

Given that the unfolded width is 4 inches, we can calculate the maximum depth by taking half of the unfolded width (since it's an isosceles triangle):
Maximum depth = 1/2 * 4 inches = 2 inches.

So, the depth of the gutter that has the greatest carrying capacity is 2 inches.