Q. 225.0( 1 Vote )

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Answer :

Given: A sphere of fixed radius r


To prove: the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is


To find: maximum volume in terms of volume of the sphere


Let R and h be the radius and height of cone respectively


Formula used:


The volume of the cone is given by,



B is center of the sphere



Using Pythagoras theorem in right triangle Δ BCD,




AB is also the radius of the sphere


AC = AB + BC






Differentiating both sides with respect to R:









To find the maximum volume, put











Squaring both sides:









If






Differentiating both sides again:






On putting



Therefore, the volume is maximum when







Hence, the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is


As the radius of sphere = r











Therefore, the Volume of cone times of volume of the sphere


OR


Given: f(x) = sin 3x – cos 3x, 0 < x < π


To find: intervals in which function f(x) is strictly increasing or decreasing


f(x) = sin 3x – cos 3x


Differentiating with respect to x:












{ sin A cos B + cos A sin B = sin (A + B)}


For interval where f(x) is increasing: f’(x) > 0




We know,


sin x > 0


0 < sin x < π and 2π < sin x < 3π


Therefore,













Therefore, f(x) is strictly increasing in



For interval where f(x) is decreasing: f’(x) < 0




We know,


sin x < 0


π < sin x < 2π and 3π < sin x < 4π


Therefore,













Therefore, f(x) is strictly decreasing in



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