Class 12 Applications Of Derivatives · Very Long Answer5 Marksrelated_rates
Class 12 Maths · Applications Of Derivatives · Very Long Answer #102
A sphere volume increases at 3 cm³/s. Find the rate of increase of its surface area when r=2 cm.
✅ Step-by-Step Solution
Final Answer: 3 cm²/s
- For radius r, V=4πr³/3 and S=4πr².
- Differentiate in time: V̇=4πr²ṙ and Ṡ=8πr ṙ.
- With V̇=3, ṙ=3/(4πr²).
- Substitute into Ṡ: Ṡ=8πr·3/(4πr²)=6/r.
- At r=2, the surface area increases at 3 cm²/s.
📋 CBSE Stepwise Marking Scheme
- [1 mark] For radius r, V=4πr³/3 and S=4πr².
- [1 mark] Differentiate in time: V̇=4πr²ṙ and Ṡ=8πr ṙ.
- [1 mark] With V̇=3, ṙ=3/(4πr²).
- [1 mark] Substitute into Ṡ: Ṡ=8πr·3/(4πr²)=6/r.
- [1 mark] At r=2, the surface area increases at 3 cm²/s.
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