The radius of a sphere is increased by 10%. Show that the volume increases by approximately 33.1%.

Step-by-step solution

Idea: Volume depends on r³, so multiplying the radius by 1.1 multiplies the volume by 1.1³.

1. Old volume = 43πr³. New radius = r + 10% of r = 1.1r.½ mark
2. New volume = 43π(1.1r)³ = 1.331 × 43πr³ (since 1.1 × 1.1 × 1.1 = 1.331).1 mark
3. Increase = (1.331 − 1) × old = 0.331 × old, i.e. 33.1%. Shown.½ mark

Check: r = 10: volume ∝ 1000; r = 11: volume ∝ 1331; (1331 − 1000) ÷ 1000 = 33.1% ✓.

Answer to write in the exam

Common mistakes that cost marks

• Answering 10% or 30% (3 × 10%).
• Working out 1.1³ as 1.33 and calling the increase 33% without the extra decimal.

How this can come in the exam

Try one yourself

The radius of a sphere is decreased by 10%. By what percentage does its volume decrease?

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