Level 3 Integration Walkthrough
2023 NCEA Level 3 Integration Question 2(e)
2023 Paper
Question
A cake factory has a container of liquid chocolate that is used in the manufacture of chocolate cakes.
The liquid chocolate is pumped out so that the rate of change of the volume remaining is proportional to the square of the volume remaining.
After one hour, the volume remaining is \(p\) litres, where \(p\) is a positive constant.
After a further hour, the volume remaining is \(\frac{4}{5}p\) litres.
Write a differential equation that models this situation, and solve it to calculate how much liquid chocolate was in the container at the start of the day, giving your answer in terms of \(p\).
First walkthrough idea
Focus to try first
Model the pumping with \(\frac{dV}{dt}=-kV^2\), solve the separable differential equation, then use the two time conditions to work back to \(V(0)\).
Step 1
Write the model
The negative sign represents the chocolate being pumped out.
Show the first step’s working
The negative sign represents the chocolate being pumped out.
Key result
\[ \frac{dV}{dt}=-kV^2 \]Walkthrough overview
What this question practises
This 2023 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Building and solving a quadratic decay model for volume.
This is Question 2(e) from the 2023 NCEA Level 3 Integration paper for AS91579 — Apply integration 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 AS91579.
Page updated .
Learning summary
Review the method, not only the answer
This walkthrough helps you practise building and solving a quadratic decay model for volume. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.