An observer stands 200 meters from the point on the ground directly beneath a hot-air…

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

An observer stands 200 meters from the point on the ground directly beneath a hot-air balloon that is rising vertically. If

\theta

is the angle of elevation of the balloon and

h

is its height above the ground so that

\tan \theta = \dfrac{h}{200}

, the balloon is ascending at a constant rate of

3\text{ m/s}

. When

\theta = \tfrac{\pi}{4}

, what is

\dfrac{d\theta}{dt}

? (a)

\dfrac{3}{800}\text{ rad/s}

(b)

\dfrac{3}{400}\text{ rad/s}

(c)

\dfrac{3\sqrt{2}}{400}\text{ rad/s}

(d)

\dfrac{3}{200}\text{ rad/s}