NCEA Calculus 2020 TWO(d - Solution & Explanation

Question

A rocket is fired vertically upwards. Its height above the launch point is given by the formula

h(t) = 4.8t^2

, where

h

is the height in metres, and

t

is the time in seconds from firing.

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An observer at point A is watching the rocket. She is at the same level as the launch point of
the rocket, and 500 m from the launch point.

Find the rate at which the angle of elevation at A of the rocket is increasing when the rocket is
480 m above the launch point.

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

Official Answer

\tan\theta=\dfrac{h}{500}

h=500\tan\theta

\dfrac{dh}{d\theta}=500\sec^2\theta=\dfrac{500}{\cos^2\theta}

t=10

\tan\theta=\dfrac{480}{500}

\theta=0.765

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

=9.6t\times\dfrac{\cos^2\theta}{500}

=96\times\dfrac{\cos^2(0.765)}{500}

=0.0999

(accept

0.1

)

Grading Criteria

Achievement (u)

Correct expression for $\dfrac{dh}{d\theta}$ .

Merit (r)

Correct expression for $\dfrac{d\theta}{dt}$ .

Excellence T1

Correct solution with correct derivatives.

Excellence T2

-

Video Explanation

NCEA Level 3 Calculus Differentiation 2020 NZQA Exam - Worked Answers by infinityplusone(starts at 28:34)

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2020 TWO(d)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2