Question

find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius cm. what is the maximum​ volume?

Answer 1

The dimensions of a right circular cylinder of maximum volume are

$5\sqrt{6}$ and $10\sqrt{3}$.

Given the radius of sphere is 15 cm.

Let's assume radius of right circular cylinder = r

height of right circular cylinder = h.

Since the cylinder is inscribed in the sphere, their centers will coincide. As a result, the height is divided into two equal portions from the sphere's center. Therefore, we can apply Pythagoras' Theorem to get the cylinder's half-height in terms of radius.

$R^2 = r^2 + (h/2)^2$.

$r^2=R^2-h^2/4$

Volume of cylinder = $\pi r^2h$

Now we can substitute value of $r^2$ in volume of cylinder.

V =  π*($R^2$ - $h^2$/4)*h

taking  R = 15 .

V = π*(225 - $h^2$/4)*h

V = π*225*h - π*$h^3$/4

For maximum volume we have to do derivative of volume = 0

derivative with respect to h.

V' = π*225 - 3*π$h^2$/4

V' = 0

π*225 - 3*π$h^2$/4 = 0

3*$h^2$/4 = 225

$h^2$ =  75*4

h = $10\sqrt{3}$

put the value of h in $r^2=R^2-h^2/4$

$r^2$ = 225 - (300/4)

$r^2$  = 150

r = $5\sqrt{6}$

To know more about right circular cylinder, here

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Complete Question here

https://www.quora.com/What-is-the-height-of-a-right-circular-cylinder-of-maximum-volume-that-can-be-inscribed-in-a-sphere-of-radius-15-cm