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Q.

The lateral edge of a regular rectangular pyramid is $' a' cm$ long . The lateral edge makes an angle $α$ with the plane of the base. The value of $α$ for which the volume of the pyramid is greatest, is

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a

$\sin^{- 1} \sqrt{\frac{2}{3}}$

b

$\frac{π}{2}$

c

$\frac{π}{3}$

d

$\cot^{- 1} \sqrt{2}$

answer is C.

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$h = a sinα & x = a cosα; x^{2} + h^{2} = a^{2} V = \frac{1}{3} y^{2} h = \frac{1}{3} 2 x^{2} h (note : 4 x^{2} = 2 y^{2} \Rightarrow y^{2} = 2 x^{2}) V (α) = \frac{2}{3} a^{2} \cos^{2} α \cdot a sinα = \frac{2}{3} a^{3} sinα \cos^{2} α now V' (α) = 0 \Rightarrow tanα = \frac{1}{\sqrt{2}}; V_{\max} = \frac{4 \sqrt{3} a^{3}}{27}$

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