NCEA Calculus 2017 THREE(d - Solution & Explanation

Question

A building has an external elevator. The elevator is rising at a constant rate of

2\ \mathrm{m\ s^{-1}}

.

Sarah is stationary, watching the elevator from a point

30\ \mathrm{m}

away from the base of the
elevator shaft.

Let the angle of elevation of the elevator floor from Sarah's eye level be

\theta

.

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Find the rate at which the angle of elevation is increasing when the elevator floor is

20\ \mathrm{m}

above Sarah’s eye level.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Official Answer

Let

h

= height above Sarah’s eye level.

\tan\theta=\dfrac{h}{30}

h=30\tan\theta

\dfrac{dh}{d\theta}=30\sec^2\theta

\dfrac{dh}{dt}=2

\dfrac{d\theta}{dt}=\dfrac{dh}{dt}\times\dfrac{d\theta}{dh}

\phantom{\dfrac{d\theta}{dt}}=2\times\dfrac{1}{30\sec^2\theta}

\phantom{\dfrac{d\theta}{dt}}=\dfrac{\cos^2\theta}{15}

At

h=20

\theta=\tan^{-1}\left(\dfrac{20}{30}\right)=0.588

\dfrac{d\theta}{dt}=\dfrac{(\cos 0.588)^2}{15}

\phantom{\dfrac{d\theta}{dt}}=0.046

radians per second

Grading Criteria

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Video Explanation

NCEA Level 3 Calculus Differentiation 2017 NZQA Exam - Worked Answers by infinityplusone(starts at 56:18)

2017 THREE(d)