A ground station equipped with a motorized satellite dish tracks a weather satellite…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A ground station equipped with a motorized satellite dish tracks a weather satellite that passes overhead along a straight line at a constant speed. Let θ(t) radians be the elevation angle of the dish above the horizontal at time t; θ increases until the satellite is directly overhead and then decreases as the satellite departs.
Assume the satellite moves at constant horizontal speed v, the closest horizontal distance between the satellite’s path and the station is d meters, and the satellite’s height above ground is H meters.
(1) Part 1: (a) Using the given quantities, write an expression for tanθ in terms of the satellite’s horizontal position s(t) measured from the point of closest approach.
(2) Part 2: (b) Explain qualitatively (no numerical calculation required) how the magnitude of dtdθ changes as the satellite approaches, passes overhead, and recedes. Your explanation should reference appropriate mathematical relationships.
(3) Part 3: (c) Discuss the sign of dtdθ before, at, and after the moment when the satellite is directly overhead, and relate this sign change to the physical motion of the dish.
(4) Part 4: (d) Identify one operational challenge the motor driving the dish might face due to the behavior of dtdθ during the satellite’s passage, and suggest a mathematical strategy the control system could use to address it.