A lamp is mounted at the top of a vertical pole on a straight
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A lamp is mounted at the top of a vertical pole on a straight, level sidewalk. The pole is 5 m tall. A pedestrian who is 1.7 m tall is walking directly away from the base of the pole at a constant speed. At a certain instant the tip of the pedestrian’s shadow lies somewhere on the sidewalk.
Throughout this problem let
• x(t) denote the pedestrian’s distance (in metres) from the base of the pole at time t seconds, and
• y(t) denote the length (in metres) of the pedestrian’s shadow at the same moment.
(a) Give a geometric argument to show that the ratio x(t)+y(t)5 equals y(t)1.7 for every t at which the quantities are defined.
(b) Without performing any calculations involving derivatives, determine whether y′(t) must be less than, equal to, or greater than x′(t) while the pedestrian is walking away from the pole. Provide a clear explanation that references the parameters of the situation.
(c) Suppose instead that the lamp were only 1.8 m above the sidewalk, still above the same 1.7 m pedestrian. Predict qualitatively how the sign and relative magnitude of y′(t) would change, if at all, and explain your reasoning.