Let's draw a vertical cross-section of the sphere and the cylinder.
The red segment connects the center of the sphere with a point on the sphere. Its length is the radius of the sphere.
The vertical segment connects the center of the sphere with the center of the bottom circle of the cylinder. The length is half the height of the cylinder.
The length of the horizontal leg of the shaded right triangle is the radius of the cylinder.
We can use the Pythagorean Theorem to write a relationship between the height and radius of the cylinder.
We can also rearrange this relationship to express the square of the radius in terms of the height.
To find the maximum volume and to find the value of h that gives this maximum, let's use a calculator to graph this function.
We begin by pushing the Y= button and typing the equation of S in the first row.
Note that we use x instead of h as the variable name, because this is what the calculator understands.
To see the graph you will need to adjust the window.
The height of the cylinder is certainly not larger than the diameter of the sphere, so let's use 0 and 16 as horizontal bounds of the graph.
Push WINDOW, change the settings, and push GRAPH.
To find the maximum point on the graph push 2nd and TRACE, then choose maximum from the menu.
The calculator will prompt you to choose a left and right bound to provide the calculator with a best guess of where the maximum might be.
Let's interpret the result we get from the calculator.
The second coordinate of the maximum point gives that the maximum volume of the cylinder is about 1238 cubic inches.
The first coordinate of the maximum point gives that the height of this cylinder with maximum volume is about h=9.24 inches.
