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. (4 points)
Make it super short and two errors
Let's denote the diameter/height of the ice cream cone as d. Since the diameter is equal to the height, we can say that d = h.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base of the cone.
Since the diameter of the cone is equal to its height, the radius of the base is half of the diameter, which is d/2.
The volume of the cone is thus V_cone = (1/3)π(d/2)^2(d) = (1/12)πd^3.
The volume of a hemisphere is given by the formula V_hemi = (2/3)πr^3, where r is the radius of the hemisphere.
Since the hemisphere is a perfect hemisphere, its radius is equal to the radius of the cone's base, which is d/2.
The volume of the hemisphere is thus V_hemi = (2/3)π(d/2)^3 = (1/6)πd^3.
Therefore, the total volume of ice cream in and on top of the cone is V_total = V_cone + V_hemi = (1/12)πd^3 + (1/6)πd^3 = (1/12 + 1/6)πd^3 = (1/12 + 2/12)πd^3 = (3/12)πd^3 = (1/4)πd^3.
Given that the volume of the hemisphere is 4 in^3, we can equate the expression for the total volume to 4 and solve for d:
(1/4)πd^3 = 4
d^3 = (4*4)/(1/4)π
d^3 = 64/(π/4)
d^3 = 64*(4/π)
d^3 = 256/π
d = (256/π)^(1/3)
To find the total volume, we can substitute this value of d into the expression for the total volume:
V_total = (1/4)π((256/π)^(1/3))^3
V_total = (1/4)π*(256/π)
V_total = 64
Therefore, the total volume of ice cream in and on top of the cone is 64 in^3.
Errors in the above solution:
1. In the initial calculation of the volume of the hemisphere, there was an error in using the formula for the volume of a sphere instead of a hemisphere.
2. In the calculation of the total volume, there was an error in the addition of the volumes of the cone and hemisphere. The correct calculation should be (1/12)πd^3 + (1/4)πd^3 = (1/12 + 3/12)πd^3 = (4/12)πd^3 = (1/3)πd^3.