Final answer:
The height and radius of the largest cylinder inscribed in a sphere of radius √5 are both 2√5. The cylinder's maximum volume is calculated with the formula V = π(5)(2√5), which simplifies to π(10√5).
Step-by-step explanation:
To find the height and radius of the largest cylinder that can be inscribed inside a sphere of radius √5, imagine a cylinder standing upright within the sphere. The cylinder's height will stretch from one end of the sphere to the other through the center, making a diameter for the sphere. This height will be twice the radius of the sphere. In our case, the sphere's radius is √5, so the height of the cylinder is 2√5.
To maximize the volume of the cylinder, we need it to have the largest possible base within the sphere. This base is a circle tangent to the sphere at all points along its edge, which occurs right at the sphere's 'equator.' The radius of the cylinder's base, therefore, will be equal to the radius of the sphere, √5. Thus, the cylinder's radius is also √5.
the volume formula for a cylinder is V = πr²h. If we were to calculate the maximum volume of the inscribed cylinder, we would have to plug in our found values for the radius and the height, which would give us the formula V = π(√5)²(2√5) = π(5)(2√5) or V = π(10√5).