4MA1/2HR Jan 2023, Question 24: Volume and Surface Area of a Cone
Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 6 marks
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The question, in short
Cone and sphere share radius r; given k times cone volume equals sphere volume, show cone total surface area equals πr²((k + √(k² + a))/k) and find constant a.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Solution · 6 marks
- 1k × (1/3)πr²h = (4/3)πr³
- 2kh = 4r
- 3h = 4r/k
- 4l² = r² + h² = r² + 16r²/k²
- 5l² = r²(k² + 16)/k²
- 6l = r√(k² + 16)/k
- 7Total surface area = πr² + πrl
- 8= πr² + πr × r√(k² + 16)/k
- 9= πr²(1 + √(k² + 16)/k)
- 10= πr²(k + √(k² + 16))/k
- 11So a = 16
Answera = 16, so the total surface area = πr²(k + √(k² + 16))/k
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