Question

Flight of an Aircraft A commuter plane takes off from the Los Angeles International Airport and touches down 30 min later at the Ontario International Airport. Let (in feet) be the altitude of the plane at time (in minutes), where . Use Rolle’s Theorem to explain why there must be at least one number with such that . Interpret your result.

Answer 1

By Rolle's Theorem, as the altitude function A(t) is continuous and
A(0) = A(30), there exists a time c between 0 and 30 where the plane's vertical speed A'(c) is zero.

Rolle's Theorem states that if a function A(t) is continuous on a closed interval (a, b), differentiable on the open interval (a, b), and A(a) = A(b), then there exists at least one number c in the interval (a, b) such that A'(c) = 0.

In the context of the aircraft's flight, A(t) represents the altitude of the plane at time t. The conditions for Rolle's Theorem are satisfied because the altitude is continuous during the flight (no abrupt changes) and the plane takes off and lands at the same airport, so A(0) = A(30).

Therefore, Rolle's Theorem ensures that at some point during the flight at (c), the rate of change of altitude A'(c) is zero. This implies that the plane momentarily levels off or changes direction, emphasizing a point of zero vertical speed during the journey.