The height of a right circular cylinder of maximum volume inscribed in...
The height of the cylinder can be found using the Pythagorean theorem. Let's assume the radius of the cylinder is r and its height is h.
We have a right triangle formed by the radius of the sphere (3), the radius of the cylinder (r), and the height of the cylinder (h). The hypotenuse of this triangle is the diameter of the sphere (which is twice the radius, or 6).
By the Pythagorean theorem, we have:
r^2 + h^2 = 6^2
r^2 + h^2 = 36
To maximize the volume of the cylinder, we need to maximize the height. Since r^2 + h^2 = 36, we can rewrite h^2 = 36 - r^2.
The volume of the cylinder is given by V = πr^2h. Substituting h^2 = 36 - r^2, we have:
V = πr^2(36 - r^2)
V = 36πr^2 - πr^4
To find the maximum volume, we can take the derivative of V with respect to r and set it equal to zero:
dV/dr = 72πr - 4πr^3 = 0
Simplifying, we have:
r(18 - r^2) = 0
This equation has two solutions: r = 0 and r = ±√18. Since the radius of a cylinder cannot be negative or zero, we discard the solution r = -√18.
Therefore, the only possible value for r is r = √18 = 3√2.
Substituting this value back into the equation r^2 + h^2 = 36, we have:
(3√2)^2 + h^2 = 36
18 + h^2 = 36
h^2 = 18
Taking the positive square root, we have:
h = √18 = 3√2
Therefore, the height of the cylinder is 3√2.
In summary, the height of the right circular cylinder of maximum volume inscribed in a sphere of radius 3 is 3√2.