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Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve this real-world problem:
A video game developer is creating a building design composed of a cylinder and a hemisphere. She wants to know the volume of her design if the volume of the cylinder is 6y. What is the volume of the whole building?
4y
12y
8y
2y
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Let the shared radius be r. For a cylinder with height 2r,
V_cyl = πr^2(2r) = 2πr^3.
A sphere is V_sph = (4/3)πr^3, so a hemisphere is half that: V_hemi = (2/3)πr^3 = (1/3)(2πr^3) = (1/3)V_cyl.
If V_cyl = 6y, then V_hemi = (1/3)(6y) = 2y, so the total volume = 6y + 2y = 8y.
Answer: 8y.
V_cyl = πr^2(2r) = 2πr^3.
A sphere is V_sph = (4/3)πr^3, so a hemisphere is half that: V_hemi = (2/3)πr^3 = (1/3)(2πr^3) = (1/3)V_cyl.
If V_cyl = 6y, then V_hemi = (1/3)(6y) = 2y, so the total volume = 6y + 2y = 8y.
Answer: 8y.
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