NCEA Calculus 2022 THREE(c - Solution & Explanation

Question

The diagram below shows the cross-section of a bowl containing water.

Loading diagram...

When the height of the water level in the bowl is

h

cm, the volume,

V

cm

^3

, of water in the bowl is
given by

V = \pi\left(\frac{3}{2}h^2 + 3h\right)

.

Water is poured into the bowl at a constant rate of

20

cm

^3

s

^{-1}

.

Find the rate, in cm s

^{-1}

, at which the height of the water level is increasing when the height of the
water level is

3

cm.

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

Official Answer

V=\pi\left(\frac{3}{2}h^2+3h\right)

\dfrac{dV}{dh}=\pi(3h+3)

\dfrac{dV}{dt}=20

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

=\dfrac{1}{\pi(3h+3)}\times 20

At

h=3,\ \dfrac{dh}{dt}=\dfrac{20}{12\pi}

=\dfrac{5}{3\pi}=0.531\ \mathrm{cm\ s^{-1}}

Grading Criteria

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Video Explanation

2022 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 56:51)

2022 THREE(c)