Metal rain gutters A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle Θ (see figure). What angle Θ maximizes the cross-sectional area of the gutter? <IMAGE>

Verified step by step guidance

1. Define the cross-sectional area A of the gutter as a function of the angle Θ. The area can be expressed as A = base * height, where the base is 3 inches and the height can be determined using trigonometric functions based on the angle Θ.
2. Express the height of the gutter in terms of the angle Θ. Since the sides are folded up at an angle Θ, the height can be represented as height = 3 * sin(Θ).
3. Substitute the expression for height into the area function to get A(Θ) = 3 * (3 * sin(Θ)) = 9 * sin(Θ).
4. To find the angle Θ that maximizes the area, take the derivative of A with respect to Θ, A'(Θ), and set it equal to zero to find critical points.
5. Analyze the critical points and the second derivative test to determine which angle Θ provides the maximum area.