2025 HSC Mathematics Extension 1
Question 12(a):
Rates of change
A sector has radius cm and angle radians. Its area remains constant at cm² while increases at radians per second. Find when cm.
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
- , so .
- , hence .
- .
- At , cm/s.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 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 sector has constant area cm². If rad/s, find when cm.
Question 2
A circle’s radius grows at cm/s. Find when cm.
Question 3
A 10 m ladder has base moving away from a wall at m/s. How fast is its top falling when the base is m from the wall?