Class 12 Applications Of Derivatives · Very Long Answer5 Marksgeometric_optimization

Class 12 Maths · Applications Of Derivatives · Very Long Answer #81

Show that a right circular cone of fixed positive volume has least curved surface area when its height is √2 times its base radius.

✅ Step-by-Step Solution

Final Answer: h=√2r.

For fixed volume V>0, V=πr²h/3 gives h=3V/(πr²), r>0.
Curved area S=πr√(r²+h²)>0, so minimize F=S²=π²r⁴+9V²/r².
F′=4π²r³−18V²/r³=0 gives 4π²r⁶=18V².
F″=12π²r²+54V²/r⁴>0, proving the unique critical point is the global minimum. Substitute V²=π²r⁴h²/9 in the critical equation to get 4r²=2h².
Since lengths are positive, h=√2r, as required.

[1 mark] For fixed volume V>0, V=πr²h/3 gives h=3V/(πr²), r>0.
[1 mark] Curved area S=πr√(r²+h²)>0, so minimize F=S²=π²r⁴+9V²/r².
[1 mark] F′=4π²r³−18V²/r³=0 gives 4π²r⁶=18V².
[1 mark] F″=12π²r²+54V²/r⁴>0, proving the unique critical point is the global minimum. Substitute V²=π²r⁴h²/9 in the critical equation to get 4r²=2h².
[1 mark] Since lengths are positive, h=√2r, as required.

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