Maxima and MinimaHard

Question

In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest -

Options

A.1/2
B.1/√3
C.1/3
D.1/4

Solution

      
Let ′a′ be the side of base.
∴    R =
⇒ a = √3 ; cos θ
volume = base area ×  height
= ×  l sin θ
f(θ) = l3 {cos2 θ sin θ}
f′(θ) = {2cos θ.(- sin2θ) + cos3θ}
= {1 - 2 tan2θ}cos3 θ
for max. volume, tan θ = 
altitude = x = l. sin θ =

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