AP Calculus AB · Differentiation definitions and fundamental rules
A drone flies horizontally at a constant altitude of 80 meters
Problem
A drone flies horizontally at a constant altitude of \(80\) meters. Its horizontal distance \(x\) from a ground observer is increasing at \(10\) meters per second. Let \(r\) be the line-of-sight distance from the observer to the drone, and let \(\theta\) be the angle of elevation (in radians) from the observer up to the drone. (a) When \(x=60\) meters, find \(\dfrac{dr}{dt}\). Include units. (b) At the same instant, find \(\dfrac{d\theta}{dt}\). Include units. (c) Interpret the sign of each answer in the context of the flight. Use \(r^2=x^2+80^2\) for the range and \(\tan\theta=\dfrac{80}{x}\) for the angle, and differentiate before substituting \(x=60\).
Hint
Pythagoras for the slant range; \(\tan\theta=80/x\) for the elevation angle. Differentiate both before plugging in \(x=60\).
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