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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMaxima-MinimaHard2 minPYQ_2019

MathematicsHardsingle choice

The maximum volume $(i n c u . m)$ of the right circular cone having slant height $3 m$ is:

Question diagram: The maximum volume i n c u . m of the right circular cone ha

Options:

Answer:

Solution:

Given $l = 3$

$\Rightarrow r^{2} + h^{2} = l^{2} = 9$

Volume, $V = \frac{1}{3} π r^{2} h$

$\Rightarrow V = \frac{1}{3} π (9 - h^{2}) h = \frac{1}{3} π (9 h - h^{3})$

$\frac{d v}{d h} = \frac{1}{3} π (9 - 3 h^{2})$

$\frac{d v}{d h} = 0 \Rightarrow h = \sqrt{3}$

$\frac{d^{2} V}{d h^{2}} = \frac{1}{3} π (- 6 h) < 0$

$∴$ at $h = \sqrt{3}$ , cone has maximum volume

$∴$ $V_{m a x} = \frac{1}{3} π (9 \sqrt{3} - 3 \sqrt{3}) = 2 \sqrt{3} π c m^{3}$

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Maxima-Minima

⏱ 2mℹ️ Source: PYQ_2019

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