Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?

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Identify the line equation y = 3x and the point (50, 0) from which we need to find the minimum distance to a point P on the line.

Express the coordinates of point P on the line as (x, 3x) since any point on the line y = 3x can be represented in this form.

Use the distance formula to express the distance D between point P(x, 3x) and the point (50, 0): D = sqrt((x - 50)^2 + (3x - 0)^2).

Simplify the distance formula: D = sqrt((x - 50)^2 + 9x^2). This simplifies to D = sqrt(10x^2 - 100x + 2500).

To find the minimum distance, minimize the expression under the square root, 10x^2 - 100x + 2500, by finding its derivative, setting it to zero, and solving for x. This will give the x-coordinate of point P that minimizes the distance.

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주요 개념

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

Distance Formula

The distance formula calculates the distance between two points in a Cartesian plane. For points (x1, y1) and (x2, y2), the distance d is given by d = √((x2 - x1)² + (y2 - y1)²). This formula is essential for determining how far the point P on the line is from the point (50, 0).

Line Equation

The equation of a line describes the relationship between x and y coordinates of points on that line. In this case, the line is given by y = 3x, which indicates that for every unit increase in x, y increases by three units. Understanding this equation helps in identifying the coordinates of point P that lies on the line.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to minimize the distance from point P on the line to the point (50, 0). This typically involves using techniques such as taking derivatives and setting them to zero to find critical points.