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. 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.
(4 points)
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Math Short Answer Rubric
To solve this problem, we can use the relationship between the formulas for the volumes of cones and spheres. The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
In this problem, the diameter of the ice cream cone is equal to its height. Let's use d to represent the diameter and h to represent the height of the cone. Since the diameter is equal to the height, we have d = h.
The radius of the base of the cone is half the diameter, so r = d/2 = h/2. Plugging this into the volume formula for a cone, we get V_cone = (1/3)π(h/2)²h = (1/12)πh³.
The volume of the hemisphere on top of the cone is given as 4 in³. Since the hemisphere is perfectly half of a sphere, its volume is half the volume of a sphere with the same radius. Let's call the radius of the hemisphere r_h. Therefore, we have (1/2)[(4/3)πr_h³] = 4.
Now, let's solve for r_h. Multiplying both sides of the equation by 2 and dividing by (4/3)π, we get r_h³ = 4 * 3/2 * (4/3)π = 8π. Taking the cube root of both sides, we have r_h = (∛8π) ≈ 1.69.
Since the radius of the hemisphere is equal to the radius of the cone's base (r_h = h/2), we can substitute this value into the formula for the volume of the cone. V_cone = (1/12)π(h³) = (1/12)π(1.69²)(1.69) = 0.225π ≈ 0.71.
Finally, the total volume of ice cream in and on top of the cone is the sum of the volume of the cone and the volume of the hemisphere. V_total = V_cone + V_hemisphere = 0.71 + 4 = 4.71 in³.
Therefore, the total volume of ice cream in and on top of the cone is approximately 4.71 cubic inches.