An observer stands on level ground watching a weather balloon rise vertically while…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

An observer stands on level ground watching a weather balloon rise vertically while drifting horizontally. The observer is located at the origin of a coordinate system. At time tt seconds, the balloon is at horizontal position x(t)x(t) meters and height y(t)y(t) meters. The angle of elevation θ(t)\theta(t) (in radians) from the observer to the balloon satisfies θ(t)=arctan⁡ ⁣(y(t)x(t)).\theta(t)=\arctan\!\left(\frac{y(t)}{x(t)}\right). At a particular instant, the balloon is 65 m above the ground and 100 m east of the observer. Measurements taken at that instant indicate that the balloon is rising at 5 m s−1^{-1} and drifting eastward at 12 m s−1^{-1}. (a) Express dθdt\dfrac{d\theta}{dt} in terms of x, y, dxdt, x,\,y,\,\dfrac{dx}{dt},\, and dydt\dfrac{dy}{dt}. Do not substitute numerical values yet. (b) Without performing any numerical calculation, determine whether dθdt\dfrac{d\theta}{dt} is positive, negative, or zero at the instant described. Provide a clear explanation that references both the analytic expression from part (a) and the physical motion of the balloon. (c) Using the data provided, evaluate dθdt\dfrac{d\theta}{dt} to four decimal places (units: rad s−1^{-1}).

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