An igloo is a shelter constructed from blocks of ice in the shape of a hemisphere. This igloo has an entrance below ground level. The outside diameter of the igloo is 12 feet. The thickness of each block of ice that was used to construct the igloo is 1.5 feet. Estimate in cubic feet the amount of space of the living area inside the igloo

The living area inside would be 190.755 ft³.

We find the volume inside the hemisphere.  The volume of a sphere is given by the formula
V = (4/3)πr³

Since this is a hemisphere, it is half of the volume:
V = 1/2(4/3)πr³ = 4/6πr³ = 2/3πr³

The diameter of the hemisphere is 12 ft.  However, there is 1.5 ft on each side that is a solid block of ice and not living space:
12 - 2(1.5) = 12-3 = 9 ft for the diameter.

The radius is half of the diameter:
9/2 = 4.5

Using this, we have 
V = 2/3πr³ = 2/3(3.14)(4.5)³ = 190.755 ft³