A rectangular sheet of metal is being stretched so that its sides remain…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A rectangular sheet of metal is being stretched so that its sides remain perpendicular at all times. One side is increasing in length at a constant rate of 4 cm/s, and the adjacent side is increasing at 3 cm/s. At the instant when the sides are 30 cm and 40 cm long, respectively, let d be the length of the diagonal of the rectangle and θ be the angle formed by the longer side and the diagonal.
(a) Write an equation relating d, the two side lengths, and any relevant geometric quantities.
(b) Determine the instantaneous rate of change of d (in cm/s) at the given instant.
(c) Find the instantaneous rate of change of θ (in rad/s) at the same instant.
(d) Explain how the differing side-length growth rates influence the behavior of d and θ over time.