A 12-ft streetlight stands at the edge of a straight road
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A 12-ft streetlight stands at the edge of a straight road. A 5-ft-tall cyclist rides directly away from the streetlight at a constant 8fts−1. The cyclist’s distance from the base of the streetlight at time t seconds is x(t) ft, and the length of the cyclist’s shadow on the road is s(t) ft.
(1) Part 1: (a) Derive an equation that relates x(t) and s(t) without using any trigonometric functions.
(2) Part 2: (b) Use the equation from part (a) to obtain an expression for s′(t) in terms of x(t) and x′(t).
(3) Part 3: (c) Find the numerical values of s′(t) and the speed of the tip of the shadow at the instant when the cyclist is 20 ft from the streetlight. Give units for each rate.
(4) Part 4: (d) Interpret the meaning of your answers in part (c) in the context of the physical situation.