MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
In a stage production, a spotlight 5 m above the stage floor shines toward a vertical movie screen located 12 m away on the same floor. An actor who is 2 m tall walks from the spotlight toward the screen at 1.4 m/s.
At time t, let x(t) be the actor’s horizontal distance from the spotlight, measured along the stage floor, and let L(t) be the height on the screen of the top of the actor’s projected image above the stage floor.
(a) Derive a relationship connecting L(t), x(t), and the fixed geometric quantities in this setting. Clearly indicate any distinct similar ratios or proportional arguments you employ.
(b) Differentiate the relationship from part (a) with respect to t to obtain an expression for dL/dt in terms of x, dx/dt, and the fixed constants.
(c) When the actor is 4 m from the spotlight, compute the instantaneous rate of change of the image height dL/dt. Include units and state whether the top of the image is moving up or down the screen.
(d) Discuss briefly how the speed of the projected image changes as the actor approaches the screen. Support your answer with reference to the expression obtained in part (b).