2021 HSC Mathematics Extension 1
Question 11(e):
Rates of change
A spherical bubble is moving up through a liquid. As it rises, the bubble gets bigger and its radius increases at the rate of .
At what rate is the volume of the bubble increasing when its radius reaches ? Express your answer in rounded to one decimal place.
How to recognise this question
Question type: rates motion
- The command and notation point to rates motion.
- The requested response is a worked solution.
How to handle it: write the geometric constraint, differentiate every changing quantity with respect to time, then substitute.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Volume , so .
- Chain rule: .
- Substitute and : .
- To one decimal place: .
- NESA: 2 marks full; 1 for using the chain rule or for obtaining from the sphere formula.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
A spherical bubble has . Find when , to one decimal place.
Question 2
If and , find to one decimal place.
Question 3
For a sphere with and , express exactly in terms of , then give one decimal place.