Two motorized platforms start simultaneously from the origin of a rectangular…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Two motorized platforms start simultaneously from the origin of a rectangular coordinate system. Platform X moves along the positive x–axis, and Platform Y moves along the positive y–axis. The platforms are connected by a flexible cable whose length L(t) is increasing at a constant rate of 5 ft/s. At the instant when Platform X is 8 ft from the origin and Platform Y is 6 ft from the origin, the vertical platform’s speed is 2 ft/s upward.
(a) Establish an equation linking x(t), y(t), and L(t) for all t.
(b) Compute the horizontal speed of Platform X at the given instant. Show all work and state units.
(c) Determine the rate at which the area of the right triangle formed by the two platforms and the origin is changing at that instant.
(d) Discuss whether the area will ever begin to decrease if the cable continues to lengthen at 5 ft/s and Platform Y maintains the 2 ft/s speed found above. Support your conclusion analytically.