Introduction
This O-Level A-Maths question tests your understanding of:
- volume of a sphere
- substitution into formulas
- rearranging equations
- working with rates and units
We follow the teacher’s exact working shown on the board, step by step.
The Question
A spherical balloon is being inflated with helium gas at a constant rate of 500 cm³ per minute.
The balloon is initially empty.
Find the radius of the balloon after one minute.
The volume of a sphere of radius \(r\) is given by:
\(V = \dfrac{4}{3}\pi r^3\)
Step-by-Step Working (Teacher’s Method)
Step 1: Find the volume after 1 minute
\(V = 500 \times 60 = 30000 \,\text{cm}^3\)
Step 2: Substitute into the sphere volume formula
\(\dfrac{4}{3}\pi r^3 = 30000\)
Step 3: Rearrange to make \(r^3\) the subject
\(r^3 = \dfrac{30000 \times 3}{4\pi} = \dfrac{22500}{\pi}\)
Step 4: Take the cube root
\(r = \sqrt[3]{\dfrac{22500}{\pi}}\)
Step 5: Final answer (to 3 significant figures)
\(r \approx 19.3 \,\text{cm}\)
✅ Final Answer
\(r = 19.3 \,\text{cm}\)
Key Concepts
| Concept | Why It Matters |
|---|---|
| Volume of a sphere | Core formula tested in A-Maths |
| Substitution | Links word problems to formulas |
| Rearranging equations | Essential algebra skill |
| Units | Must stay consistent throughout |
Tips for Students
- Always calculate volume before using formulas
- Write the sphere formula clearly
- Rearrange carefully before substituting numbers
- Give answers to the correct number of significant figures
For Parents
This question develops:
- formula application skills
- algebraic confidence
- exam accuracy under pressure
These are high-frequency skills tested in O-Level A-Maths.