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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
The lateral edge of a regular rectangular pyramid is $'a'$ cm long . The lateral edge makes an angle $\alpha$ with the plane of the base. The value of $\alpha$ for which the volume of the pyramid is greatest, is
  • A
    $\frac{\pi }{4}$
  • B
    $sin^{-1}\sqrt {\frac{2}{3}} $
  • ✓
    $cot^{-1} \sqrt {2} $
  • D
    $\frac{\pi }{3}$
✓

Answer

Correct option: C.
$cot^{-1} \sqrt {2} $
c
$h = a sin \alpha$ & $x = a cos \alpha $ ;

$x^2 + h^2 = a^2$

$V = \frac{1}{3}y^2 h = \frac{1}{3} 2 x^2 h$   (note $4x^2 = 2y^2 ==> y^2 = 2x^2$)

$V (\alpha ) = \frac{2}{3}a^2 cos^2 \alpha . a sin \alpha = \frac{2}{3}a^3 sin \alpha cos^2 \alpha $

now $ V' (\alpha ) = 0$ ==> $tan \alpha = \frac{1}{{\sqrt 2 }}$ ;

$V_{max} =\frac{{4\,\sqrt 3 \,\,{a^3}}}{{27}}$ 

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