Differential Equations Worked Solution – 2022 NCEA Level 3 Q2(e) (AS91579)

Question

A cylindrical tank of height \(150\text{ cm}\) is originally full of oil.

The height \(h\), in cm, of the oil left in the tank after it has been leaking for \(t\) minutes can be modelled by

\[ \frac{dh}{dt}=-\frac{1}{4}\sqrt{(h-6)^3}. \]

Find how long it takes for the oil to be \(15\text{ cm}\) above the bottom of the tank.

Initially the tank is full, so \(h=150\) when \(t=0\).

First walkthrough idea

Focus to try first

Separate the variables carefully, integrate, use the full-tank starting condition, then substitute the target height.

Step 1

Separate the variables

Move all the \(h\)-terms to one side and the \(t\)-terms to the other.

Show the first step’s working

\frac{dh}{dt}=-\frac{1}{4}\sqrt{(h-6)^3} \] \[ \frac{1}{\sqrt{(h-6)^3}}\,dh=-\frac{1}{4}\,dt \] \[ (h-6)^{-3/2}\,dh=-\frac{1}{4}\,dt

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Solving a separable leakage model for the required time.

This is Question 2(e) from the 2022 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 9 August 2026.

Learning summary

Review the method, not only the answer

This walkthrough helps you practise solving a separable leakage model for the required time. 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.

Continue practising

← Question 2(d) Question 3(a) →