Level 3 Integration Walkthrough
2019 NCEA Level 3 Integration Question 3(e)
2019 Paper
Question
An inverted right pyramid has a square base of side length \(0.9\) metres and a height of \(1.5\) metres. The pyramid is initially filled with water to a depth of \(1\) metre.
The energy required to pump water out of a tank of height \(H\) is given by \(E=9800\int_{H-d}^{H}(H-h)A(h)\,dh\), where \(E\) is the energy in joules, \(d\) is the initial depth of the water in the tank, \(h\) is the depth of the water in the tank at any instant, and \(A(h)\) is the area of the surface of the water at this instant.
Find the energy required to pump the water out of the tank shown.
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Focus to try first
Use similar triangles to express the water-surface area \(A(h)\), then substitute it into the supplied pumping-energy integral.
Step 1
Set the limits in the given formula
Here \(H=1.5\) m and the initial depth is \(d=1\) m, so \(H-d=0.5\).
Show the first step’s working
Walkthrough overview
What this question practises
This 2019 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Pumping energy with similar triangles and a definite integral.
This is Question 3(e) from the 2019 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.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise pumping energy with similar triangles and a definite integral. 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.