Class 12 Applications Of Derivatives · Very Long Answer5 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #73

An open-top box with square base is formed from c² square units of cardboard, c>0, with no waste and negligible thickness. Show its greatest volume is c³/(6√3).

✅ Step-by-Step Solution

Final Answer: Vmax=c³/(6√3); base side c/√3, height c/(2√3).

Let base side x>0 and height h>0. Material area gives x²+4xh=c², hence h=(c²−x²)/(4x), with 0<x<c.
V=x²h=(c²x−x³)/4.
V′=(c²−3x²)/4=0 gives x=c/√3.
The derivative is positive before and negative after this point; the limiting endpoint volumes are zero, so it is the global maximum.
h=c/(2√3) and Vmax=(c²/3)(c/(2√3))=c³/(6√3).

📋 CBSE Stepwise Marking Scheme

[1 mark] Let base side x>0 and height h>0. Material area gives x²+4xh=c², hence h=(c²−x²)/(4x), with 0<x<c.
[1 mark] V=x²h=(c²x−x³)/4.
[1 mark] V′=(c²−3x²)/4=0 gives x=c/√3.
[1 mark] The derivative is positive before and negative after this point; the limiting endpoint volumes are zero, so it is the global maximum.
[1 mark] h=c/(2√3) and Vmax=(c²/3)(c/(2√3))=c³/(6√3).

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Class 12 Maths · Applications Of Derivatives · Very Long Answer #73

✅ Step-by-Step Solution

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Master Chapter 6: Applications Of Derivatives

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