Level 3 Differentiation Walkthrough
2018 NCEA Level 3 Differentiation Question 2(e)
2018 Paper
Question
A water tank is in the shape of an inverted right-circular cone. The height of the cone is \(200\text{ cm}\) and its radius is \(80\text{ cm}\).
The tank is being filled with water at a rate of \(150\text{ cm}^3\) per second. At what rate will the circular surface area of the water be increasing when the depth is \(125\text{ cm}\)?
You must use calculus and show any derivatives that you need to find when solving this problem.
First walkthrough idea
Focus to try first
Use similar triangles to write both volume and surface area in terms of the single variable \(h\), then connect their rates.
Step 1
Use similar triangles
The water cone and tank cone have the same radius-to-height ratio.
Show the first step’s working
Walkthrough overview
What this question practises
This 2018 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Related rates for surface area in a conical tank.
This is Question 2(e) from the 2018 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 for surface area in a conical tank. 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
- All 2018 Differentiation walkthroughs
- All AS91578 Differentiation years
- Practise more questions using this skill: Related rates