4MA1/1H Jan 2021, Question 17: Stationary Points and Turning Points
Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 5 marks
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The question, in short
A shape combines a hemisphere on a cylinder, both radius x cm, cylinder height (20 − 4x) cm. Given V = (1/3)πy, find the largest possible y, showing full algebra.
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Solution · 5 marks
- 1V = (2/3)πx³ + πx²(20 − 4x) = 20πx² − (10/3)πx³
- 2V = (1/3)πy, so y = 60x² − 10x³
- 3dy/dx = 120x − 30x² = 0 gives x = 0 or x = 4
- 4d²y/dx² = 120 − 60x = −120 at x = 4, so this is a maximum
- 5y = 60 × 16 − 10 × 64 = 960 − 640 = 320
Answer320
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