Answer:
128π/3 cubic units
Step-by-step explanation:
You want the volume of a cylinder inscribed in a triangular prism that has all edges 8 units long.
Base
The base of the prism is an equilateral triangle with side length 8. Such a triangle has an altitude of ...
8(√3/2) = 4√3
The centroid of the triangle is also the center of the incircle. That is located 1/3 of the length of the altitude from each side. That distance is the radius of the inscribed cylinder.
r = (4√3)/3 = 4/√3
Then the area of the base of the cylinder is ...
A = πr² = π(4/√3)² = 16π/3
Volume
The height of the cylinder is 8 units, so the volume is ...
V = Bh
V = (16π/3)(8) = 128π/3
The volume of the inscribed cylinder is 128π/3 cubic units.
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