NCEA Calculus 2021 TWO(d - Solution & Explanation

Question

The volume of a spherical balloon is increasing at a constant rate of

60\ \mathrm{cm}^3

per second.

Find the rate of increase of the radius when the radius is

15\ \mathrm{cm}

.

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

Official Answer

\dfrac{dV}{dt}=60

V=\dfrac{4}{3}\pi r^3

\dfrac{dV}{dr}=4\pi r^2

\dfrac{dr}{dt}=\dfrac{dV}{dt}\times\dfrac{dr}{dV}

=\dfrac{60}{4\pi r^2}

=\dfrac{15}{\pi r^2}

r=15\Rightarrow\dfrac{dr}{dt}=\dfrac{15}{\pi 15^2}

=\dfrac{1}{15\pi}\;(=0.0212)\;\text{cm s}^{-1}

Grading Criteria

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Video Explanation

NCEA Level 3 Calculus Differentiation 2021 NZQA Exam - Worked Answers by infinityplusone(starts at 41:50)

2021 TWO(d)