Rates of change — Surface-area recognition

Question

Explain briefly why for a sphere, and use it to find when .

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. is the surface area of the sphere, equal to .
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: .

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

Verification: Related rates for expanding spheres divide by surface area.

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