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2020 HSC Mathematics Extension 1

Question 10:
Rates of change

1 markrates motion

The quantities , and are connected by the related rates

,\quad ,\quad ,

where , and are non-zero constants.

Which of the following statements is true?

  1. is increasing and is increasing
  2. is increasing and is decreasing
  3. is decreasing and is increasing
  4. is decreasing and is decreasing

How to recognise this question

Question type: rates motion

  • The command and notation point to rates motion.
  • The requested response is a multiple choice.

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

  1. Given for any non-zero .
  2. Then , so is increasing.
  3. From and , we get , so is increasing.
  4. Both rates of change are positive.
  5. Answer: A.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark 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

Suppose , , with . Then

  1. decreasing and decreasing
  2. increasing and increasing
  3. decreasing and increasing
  4. increasing and decreasing
See the step-by-step answer guide →

Question 2

Suppose , , with . Then

  1. increasing and decreasing
  2. increasing and increasing
  3. decreasing and decreasing
  4. decreasing and increasing
See the step-by-step answer guide →

Question 3

If and with , then equals

See the step-by-step answer guide →