Amachine part consists of a half sphere and a cylinder, as shown in the figure. the total volume of the part is
cubic inches.

A machine consist of a half sphere and a cylinder, so the total volume is going to be given by:

Volume = Cylinder [in^3] + Half-sphere [in^3]

The volume of a cylinder is given by:

Vcylinder = πr^2h, where:

h: Height of the cylinder [in]

r: radius of the cylinder [in]

The volume of a sphere is given by:

Vsphere= 4/3πr^3, where:

r: Radius of the sphere [in]

Given that the machine consistsof a half sphere, then the radius of the half sphere is:

Vhalf-sphere= 4/6πr^3.

Therefore, the total volume is:

Total volume = Vhalf-sphere + Vcylinder

Total volume = 4/6πr^3 + πr^2h [inches^3]

let's first notice, the center of the hyperbola is at the origin, since the focus points are equidistantly 3 units away from the origin, that also means that the distance "c" from the center to either focus is 3.

it should also be notable that, the traverse axis for this hyperbola is vertical, or namely over the y-axis and therefore, the positive fraction will be the one with y-variable.

the traverse "a" component or distance from the center to either vertex is not 2, is more than 2, so we can safely put it at √5 ≈ 2.24.