Question

The maximum volume of a cylinder inscribed in a sphere of radius 7 is​

Answer 1

Step-by-step explanation:

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Answer 2

The maximum volume of cylinder is 414.76 cu. unit

Step-by-step explanation:

A cylinder inscribed in a sphere of radius 7.

Let radius of cylinder be "r" and height "h"

In ΔOAB, ∠OAB = 90°

$OA^2+AB^2=OB^2$     By pythagoreous theorem

$(\frac{h}{2})^2+r^2=7^2$

$r^2=49-\dfrac{h^2}{4}$

Volume of cylinder, $V=\pi r^h$

$V(h)=\pi (49-\dfrac{h^2}{4})h$

Derivative w.r.t h

$V'(h)=\pi (49-\dfrac{3h^2}{4})$

For critical point, V'(h)=0

$\pi (49-\dfrac{3h^2}{4})=0$

$h=\dfrac{7}{\sqrt3}$

$r=7\sqrt{\dfrac{2}{3}}$

Using double derivative test,

$V''(h)=\dfrac{6\pi h}{4}$

$V''(\dfrac{7}{\sqrt3})>0$ (max)

$V_{max}=\pi \times 49\times \dfrac{2}{3}\times \dfrac{7}{\sqrt3}$

$V_{max}=414.76\text{ cu. unit}$

The maximum volume of cylinder is 414.76 cu. unit

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