Level 3 Differentiation Walkthrough

2022 NCEA Level 3 Differentiation Question 3(c)

2022 Paper - Related rates for a bowl of water

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Question

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 \(20\text{ cm}^3\text{ s}^{-1}\).

Find the rate, in \(\text{cm s}^{-1}\), at which the height of the water level is increasing when \(h=3\text{ cm}\).

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

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First walkthrough idea

Tip to try first

The pouring rate gives you \(\frac{dV}{dt}\), not \(\frac{dh}{dt}\).

Step 1

Identify the given rate

Reveal the explanation and working for this step.

Show the first step’s working

Read the given volume rate as a derivative with respect to time.

Key result

\[ \frac{dV}{dt}=20 \]

The question gives the rate at which volume changes: \(\frac{dV}{dt}=20\).

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Related rates using volume and height.

This is Question 3(c) from the 2022 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise related rates using volume and height. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Differentiate with respect to time consistently, then include the correct units and contextual interpretation.

Continue practising

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