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>

검증된 단계별 안내

First, understand that the cross-sectional area of the gutter is a trapezoid. The base is fixed at 3 inches, and the two sides are 3 inches each, folded at an angle Θ. The goal is to express the area as a function of Θ.

Next, express the height of the trapezoid in terms of Θ. The height can be found using trigonometry: it is the vertical component of the side, which is 3 * sin(Θ).

Now, express the top width of the trapezoid. The top width is the horizontal component of the two sides, which is 3 * cos(Θ) for each side. Therefore, the total top width is 3 + 2 * 3 * cos(Θ).

Write the formula for the area of the trapezoid: A(Θ) = (1/2) * (base + top width) * height. Substitute the expressions for the base, top width, and height in terms of Θ.

To find the angle Θ that maximizes the area, take the derivative of A(Θ) with respect to Θ, set it equal to zero, and solve for Θ. This will give the critical points. Use the second derivative test or analyze the critical points to determine which angle maximizes the area.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cross-Sectional Area

The cross-sectional area of a shape is the area of a slice taken perpendicular to its length. In the context of the rain gutter, it refers to the area formed by the base and the sides of the gutter when folded. Maximizing this area is crucial for ensuring the gutter can effectively channel water.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to determine the angle Θ that maximizes the cross-sectional area of the gutter. This typically involves using techniques such as taking derivatives and applying critical point analysis.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In this scenario, the angle Θ will influence the height of the sides of the gutter, which in turn affects the cross-sectional area. Understanding these relationships is essential for setting up the optimization problem.