Applications Of Derivatives Question 65
Question: A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle $ \theta $ with the horizontal. The value of $ \theta $ for which the height of G, the mid-point of the rod above the peg is minimum, is
Options:
A) $ 15{}^\circ $
B) $ 30{}^\circ $
C) $ 60{}^\circ $
D) $ 75{}^\circ $
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Answer:
Correct Answer: C
Solution:
[c] We have $ AC=\sec \theta ,AG=8 $
$ \therefore CG=8-\sec \theta $ (C being the peg). But $ u=CG\sin \theta =(8-sec\theta )sin\theta $
$ u=8\sin \theta -\tan \theta $
$ \frac{du}{d\theta }=8\cos \theta -{{\sec }^{2}}\theta , $
$ \frac{d^{2}u}{d{{\theta }^{2}}}=-8\sin \theta -2{{\sec }^{2}}\theta \tan \theta $
$ \frac{du}{d\theta }=0, $ when $ {{\cos }^{3}}\theta =\frac{1}{8},\cos \theta =\frac{1}{2}, $
$ \frac{d^{2}u}{d{{\theta }^{2}}}>0(at\theta =60{}^\circ ),\therefore \theta =60{}^\circ $
BETA
