Question

What is the volume of a spherical dome with radius 5.5ft and height 3.5ft?

Answer 1

Hmmm. Exactly what is height? Is this a truncated sphere?

Answer 2

The way I read this:
A spherical cap is cut off giving a truncated sphere as bobpursley suspected.

Look at

http://en.wikipedia.org/wiki/Spherical_cap

to see how we can find the volume of that cap.
Volume = (πh/6)(3a^2 + h^2) , where a is the radius of the circular base of the cap, and h is its height.

To find a, look at a cross-section, and
a^2 + 2^2 = 5.5^2
a^2 = 26.25

and our h = 3.5

volume of cap = (3.5π/6)(3(26.25) + 3.5^2)

volume of whole sphere = (4π/3)(5.5^3)

subtract the volume of the cap from the volume of the sphere.

I will let you do the rest of the button-pushing.