Rate Of Change Of Quantities

Concept: Calculus - Application of Derivatives (Rates of Change) and Coordinate Geometry.

Step 1: Define the coordinates of the moving point A. Point A lies on the line . Let the x-coordinate of point A be . Then, its y-coordinate is . Thus, .

Step 2: Determine the rate of change of coordinate . We are given that point A is moving along the line at a speed of 2 units/second. This speed is the rate of change of the distance from the origin to . Distance . The rate of change is . Differentiating with respect to time : . Therefore, . Let's call this Equation (i).

Step 3: Set up the formula for the area of . The area ( ) of a triangle with vertices is given by . Substitute the vertices , , and : .

Step 4: Simplify the area expression. Expand the terms inside the absolute value: . Since A is moving along the line from the origin upwards (indicated by positive speed), increases, making positive for sufficiently large . For the rate of increase, we differentiate the expression: .

Step 5: Differentiate the area with respect to time to find the final rate. Differentiate with respect to time : . Substitute the value of from Equation (i): . $ $