Let be a right triangle in the plane. The angle at vertex is the right angle, the vertex is fixed at the origin, and the vertex lies on the parabola . At time , the vertex is at the point and moves up the -axis at a constant rate of 2 centimeters per second. What is the rate of change of the area of the triangle at time seconds?
Here’s a graph of the setup:
We are given that the vertex moves upward along the -axis at a constant rate of 2 centimeters per second, this means,
Then, we compute the area of the triangle, call it , in terms of and ,
(Where we used the product rule to differentiate with respect to , recalling that both and are functions of as well.) We are then given the formula for with respect to ,
So, when we have (see the comments for an explanation of how we calculated ) which implies . Therefore,