Rates of change — Exact before rounding

Question

For a sphere with and , express exactly in terms of , then give one decimal place.

This practice question was inspired by Question 11(e) in the 2021 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..
State the final answer clearly in the form the question requested. (one d.p.).

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

Verification: Exact form or .

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