Maximum area A line segment of length 10 joins the points (0, p) and (q, 0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.

검증된 단계별 안내

First, understand that the line segment forms a right triangle with the x-axis and y-axis. The vertices of the triangle are (0, 0), (0, p), and (q, 0). The area of a triangle is given by the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Here, the base is q and the height is p.

Next, use the distance formula to express the relationship between p and q. The length of the line segment is 10, so we have: \( \sqrt{p^2 + q^2} = 10 \). This can be rewritten as \( p^2 + q^2 = 100 \).

Express the area of the triangle in terms of one variable using the constraint \( p^2 + q^2 = 100 \). The area \( A \) can be written as \( A = \frac{1}{2}pq \).

To maximize the area, substitute \( q = \sqrt{100 - p^2} \) into the area formula: \( A = \frac{1}{2}p\sqrt{100 - p^2} \).

Differentiate the area function with respect to p, set the derivative equal to zero, and solve for p to find the critical points. Then, use the second derivative test or analyze the critical points to determine which values of p and q maximize the area.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

주요 개념

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

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * base * height. In this context, the base and height are determined by the coordinates of the points where the line segment intersects the axes. Understanding how to express the area in terms of the variables p and q is essential for maximizing it.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. This typically requires taking the derivative of the function, setting it to zero to find critical points, and using the second derivative test to determine whether these points correspond to maxima or minima. In this problem, we will optimize the area function with respect to p and q.

Constraints

Constraints are conditions that must be satisfied in optimization problems. In this case, the length of the line segment is fixed at 10, which creates a relationship between p and q. Recognizing and incorporating this constraint is crucial for correctly formulating the area function and ensuring that the solution adheres to the problem's requirements.