To find the volume of the solid, we need to subtract the volume of the hemispherical hole from the volume of the cylinder.
The diameter of the hemispherical hole is equal to the thickness of the rim plus the diameter of the cylinder, which is 3 + 2r, where r is the radius of the cylinder.
Using the Pythagorean theorem, we can find the radius of the cylinder:
r^2 + (12/2)^2 = (11.5/2)^2
r^2 + 36 = 33.0625
r^2 = 33.0625 - 36
r^2 = -2.9375
r = sqrt(-2.9375) (ignoring the imaginary solution)
r ≈ 1.71
Therefore, the diameter of the hemispherical hole is 3 + 2r ≈ 6.42 inches.
The height of the cylinder is 12 inches, so its volume is:
V_cylinder = πr^2h
V_cylinder = π(1.71)^2(12)
V_cylinder ≈ 116.71 cubic inches
The volume of the hemispherical hole is:
V_hemisphere = (2/3)π(0.5d/2)^3
V_hemisphere = (2/3)π(0.5(6.42)/2)^3
V_hemisphere ≈ 18.67 cubic inches
Therefore, the volume of the solid is:
V_solid = V_cylinder - V_hemisphere
V_solid ≈ 116.71 - 18.67
V_solid ≈ 98.04 cubic inches
So the volume of the solid is approximately 98.04 cubic inches.