An outdoor event uses a telescoping light pole whose height can be varied while the…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
An outdoor event uses a telescoping light pole whose height can be varied while the event is in progress. The pole’s light source is located at its top.
At a particular moment, the following data are recorded:
• Height of the pole: 4.0 m and increasing at 0.050 m/s.
• A photographer 1.70 m tall is walking directly away from the base of the pole at 1.0 m/s.
• The photographer is 8.0 m from the pole’s base.
Let x(t) be the horizontal distance between the photographer and the pole, and s(t) the length of the photographer’s shadow.
(a) Establish an equation relating x, s, and the current height h of the pole. Explicitly indicate how the changing pole height enters your equation.
(b) Differentiate the equation from part (a) with respect to t to obtain a formula that includes dx/dt, ds/dt, and dh/dt. Show each differentiation step.
(c) Using the recorded data, calculate ds/dt, the rate at which the photographer’s shadow length is changing at that moment.
(d) Determine the speed of the shadow’s tip along the ground at that same instant and comment on whether it exceeds the photographer’s own walking speed.