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# Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius 5(sqrt 3)cm is 500 pi cm^3

6 years ago

putting the values give maximum volume of the cylinder which can be inscribed in a sphere of radius 5(sqrt 3)cm is 500 pi cm^3
Arun Kumar IIT Delhi
6 years ago
Hi
$\\r_{cyl}=rcos\theta \\h_{cyl}=2rsin\theta \\v_{cyl}=\pi (rcos\theta)^22rsin\theta \\lnv_{cyl}=ln(cons)+2lncos\theta+lnsin\theta=0 \\=>0=-2sin\theta/cos\theta+cos\theta/sin\theta \\=>cos^2\theta=2sin^2\theta \\=>sin\theta=1/\sqrt3 \\r_{cyl}=r*\sqrt{2/3} \\h_{cyl}=r*1/\sqrt3$

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
Priya
11 Points
4 years ago
Draw a sphere inside the sphere draw a cylinder .Draw a line through the centre if the cylinder (diagonal)
If that length is AB then AB=2R
AC (diameter of the circlular portion if the cylinder)=2r and BC(height)=h.
AB2=AC2+BC2
4R2=4r2-h2/4
Volume = pi r2h
substitute for r2.
differentiate it.put dv/dh=0.
u get h=2R/root(3).
put that value in vol eq to get the required ans
hope u understood..!