A building is in the form of a right circular cylinder surmounted by a
hemispherical dome both having the same base radii. The base diameter of the
dome is equal to of the total height of the building. Find the height of the
building, if it contains of air.

Let the radius of the base of the cylinder and the dome be . The total height of the building consists of the height of the cylinder and the radius of the dome. Given that the height of the dome is equal to its radius, we can express the total height of the building as:

The problem states that the base diameter of the dome is equal to of the total height of the building. Thus, we can write:

Substituting gives us:

Simplifying this equation results in:

Now we substitute back into the equation for :

The volume of the building, which is made up of the cylinder and the hemisphere on top, is:

Substituting :

This is equal to the given volume:

To convert to an improper fraction:

Setting the two volumes equal gives us:

Next, solving for :

Taking the cube root to find :

Now substituting back to find :

Finally, substituting back into :

To calculate the values, use approximations for and compute and numerically. Thus we get the height of the building.