4PM1/01 Jun 2022, Question 8: Connected Rates of Change and Small Changes
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 7 marks
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The question, in short
Sphere volume is 500 cm³; find its radius to 3 s.f., then use calculus to estimate the radius increase when surface area rises by 20 cm², to 2 s.f.
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Part (a) · 2 marks
Calculate the sphere's radius in cm, giving the answer to 3 significant figures.
- 1(4/3)πr³ = 500
- 2r³ = (3 × 500)/(4π) = 119.4…
- 3r = 4.92 cm
Answerr = 4.92 cm
Part (b) · 5 marks
Using calculus, estimate the increase in radius in cm, to 2 significant figures, for a 20 cm² surface area increase.
- 1δA = 20
- 2A = 4πr² ⇒ dA/dr = 8πr
- 3δr ≈ (dr/dA)δA = 20/(8πr)
- 4r = 4.92: δr ≈ 20/(8π × 4.92) = 0.16 cm
Answer0.16 cm
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