MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
In a manufacturing process, a rectangular frame must maintain a constant area while its length and width vary with time.
At a certain instant the length is decreasing. The manager claims that the frame’s perimeter must therefore also be decreasing at that moment.
(1) Explain, without performing any numerical calculation, why the manager’s claim may be incorrect. Use reasoning based on relationships among the instantaneous rates of change of area, length, width, and perimeter.
(2) Describe a specific scenario (qualitative or with hypothetical numbers) in which the length is decreasing yet the perimeter is increasing. Justify that your example satisfies the constant-area condition.