Level 3 Integration Walkthrough
2024 NCEA Level 3 Integration Question 3(e)
2024 Paper
Question
A teacher makes a cup of coffee at the start of the interval. The teacher leaves the cup of coffee in the staff room where the temperature is \(18^\circ\text{C}\).
After \(30\) minutes, the temperature of the cup of coffee is \(50^\circ\text{C}\).
When the teacher returns again, after a further one hour, the cup of coffee has cooled down to a temperature of \(30^\circ\text{C}\).
The rate at which the temperature of the cup of coffee changes at any instant is proportional to the difference between the temperature of the cup of coffee, \(N\), and the temperature of the room.
Write a differential equation that models this situation, and then solve it to calculate the temperature of the cup of coffee when it was made.
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Hint to try first
The cooling law compares the coffee temperature to the room temperature \(18^\circ\text{C}\), so the model is \(\frac{dN}{dt}=-k(N-18)\).
Step 1
Write the model
The negative sign captures cooling toward the room temperature.
Show the first step’s working
The negative sign captures cooling toward the room temperature.
Key result
\[ \frac{dN}{dt}=-k(N-18) \]Walkthrough overview
What this question practises
This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Building and solving a Newton's law of cooling model.
This is Question 3(e) from the 2024 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 Newton's law of cooling model. 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.