A hot-air balloon is rising vertically at a constant rate

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A hot-air balloon is rising vertically at a constant rate. An observer stands on level ground 120 m from the point on the ground directly beneath the balloon. Let θ(t)\theta(t) be the angle of elevation from the observer to the balloon, measured in radians, at time tt seconds, and let h(t)h(t) be the balloon’s height in meters above ground level. (a) Write an explicit equation relating hh and θ\theta. (b) When θ=π6\theta=\dfrac{\pi}{6}, the balloon is rising at 4.0 m/s4.0 \text{ m/s}. Calculate dθdt\dfrac{d\theta}{dt} at that moment and state appropriate units. (c) How fast is the angle of elevation changing when θ=π3\theta=\dfrac{\pi}{3} if the balloon continues to rise at 4.0 m/s4.0 \text{ m/s}? Interpret the sign of your answer.

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