A right circular cylinder in inscribed in a sphere or radius r. Find the largest volume of the cylinder.?

Question

Its an opitimization problem

Answer 1

Let half height of cylinder = h. Then radius of cylinder is Sqrt (r^2 - h^2), and the volume of the cylinder is:

V = pi (r^2 - h^2) 2h
= 2 pi r^2 h - 2 pi h^3

Differentiate with respect to h, and we get:

2 pi r^2 - 6 pi h^2

Setting this to 0 for max/min, we have

2 r^2 - 6 h^2 = 0

h = Sqrt (1/3) r

Since we know that

V = 2 pi r^2 h - 2 pi h^3

We have:

V = 2 Sqrt (1/3) pi r^3 - 2/3 Sqr (1/3) pi r^3
= 4/3 Sqrt (1/3) pi r^3
= 0.577 4/3 pi r^3

That is, the maximum volume of the cylinder is 0.577 of the sphere.

Answer 2

enable the radius of the cylinder be a and ideal be b. Then a = r(a million-b/h). the quantity of the cylinder is V = pi a^2 b = pi r^2 (a million-(b/h))^2 b dV/db = 0 ==> b=h or b=h/3. the muse b=h is dropped once you think approximately that it delivers 0 volume. as a effect b=h/3. A = r(a million-(h/3)/h) = 2r/3. V = pi*(2r/3)^2 * h/3 = (4/27) pi r^2 h.

Answer 3

when the diameter of the cylinder and the height are equal