Rates of change — Related rates sphere show-that

Question

Given and , show that .

This practice question was inspired by Question 11(f) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: write the geometric constraint, differentiate every changing quantity with respect to time, then substitute..
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Answer: as required.

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Simplifying yields the stated form.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.