Mathematics

Problem

An observer stands on level ground at point

O

. A hot-air balloon rises vertically from point

A

, which is

50\text{ m}

horizontally from

O

. At time

t

, let

y

denote the balloon’s height above the ground, and let

\theta

denote the angle of elevation of the balloon as seen from

O

. Suppose the balloon is ascending at a constant rate of

4\text{ m/s}

. (a) Write a trigonometric equation that relates

y

and

\theta

. Clearly identify all variables. (b) Differentiate your equation from part (a) with respect to time to obtain a relationship between

\dfrac{dy}{dt}

and

\dfrac{d\theta}{dt}

. Then determine

\dfrac{d\theta}{dt}

(in radians per second) when

y=30\text{ m}

. (c) At the instant when

y=30\text{ m}

, calculate the rate at which the straight-line distance between the observer and the balloon is changing. Indicate whether that distance is increasing or decreasing.

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