Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 1(c)

2017 Paper

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Question

An object's acceleration is modelled by the function

\[ a(t)=1.2\sqrt{t}, \]

where \(a\) is the acceleration of the object, in \(\text{m s}^{-2}\), and \(t\) is the time in seconds since the start of the object's motion.

If the object had a velocity of \(7\text{ m s}^{-1}\) after 4 seconds, how far did it travel in the first 9 seconds of motion?

You must use calculus and show the results of any integration needed to solve the problem.

First walkthrough idea

Focus to try first

Integrate acceleration to obtain velocity, use the velocity at four seconds, then integrate velocity over the first nine seconds.

Step 1

Write acceleration as a power

Use \(\sqrt t=t^{1/2}\) before integrating.

Show the first step’s working
\[ a(t)=1.2t^{1/2}. \]

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Building velocity and displacement from acceleration.

This is Question 1(c) from the 2017 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 building velocity and displacement from acceleration. 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 1(b) Question 1(d) →