Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in^3. The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.
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The volume of a sphere is given by the formula V = 4/3πr³, where r is the radius of the sphere. The volume of a cone is given by the formula V = 1/3πr²h, where r is the radius of the base of the cone and h is the height of the cone.
Given that the hemisphere of ice cream on top of the cone has a volume of 4 in³, we can find the radius of the hemisphere by setting the volume of a sphere equal to 2*4 in³ (since a hemisphere is half of a sphere) and solving for r:
2*4 = 4/3πr³
8 = 4/3πr³
r³ = 8*3/4π
r³ = 6/π
r = (6/π)^(1/3)
Since the diameter of the ice cream cone is equal to its height, the radius of the base of the cone is equal to the radius of the hemisphere, and the height of the cone is twice the radius of the hemisphere. Therefore, the volume of the cone is:
V = 1/3πr²h
V = 1/3π[(6/π)^(1/3)]²*2*(6/π)^(1/3)
V = 1/3π*(6/π)^(2/3)*2*(6/π)^(1/3)
V = 2/3π*(6/π)
V = 4 in³
Therefore, the total volume of ice cream in and on top of the cone is 4 in³ (from the hemisphere) + 4 in³ (from the cone) = 8 in³.
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