2024 HSC Mathematics Extension 1
Question 11(f):
Rates of change
The volume of a sphere of radius cm is given by , and the volume of the sphere is increasing at a rate of .
Show that the rate of increase of the radius is given by .
How to recognise this question
Question type: rates motion
- The command and notation point to rates motion.
- The requested response is a show that.
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
- Differentiate with respect to : .
- Chain rule: .
- This is the required rate of increase of the radius.
- NESA 11(f): 2 marks for the correct solution; 1 mark for correctly applying the chain rule to find an expression for .
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
Given and , show that .
Question 2
If and , find in terms of .
Question 3
Explain briefly why for a sphere, and use it to find when .