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)
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height.
Since the diameter of the cone is equal to its height, let's call both of them x. Therefore, the radius of the cone is (1/2)x.
The volume of the cone is then Vcone = (1/3)π((1/2)x)^2(x) = (1/12)πx^3.
The volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius of the sphere.
Since the diameter of the hemisphere is also x, the radius of the hemisphere is (1/2)x.
The volume of the hemisphere is then Vhemisphere = (2/3)π((1/2)x)^3 = (1/6)πx^3.
To find the total volume of the ice cream in and on top of the cone, we add the volumes of the cone and the hemisphere.
Total volume = Vcone + Vhemisphere = (1/12)πx^3 + (1/6)πx^3 = (1/12 + 1/6)πx^3 = (1/4)πx^3.
We are given that the volume of the hemisphere is 4 in^3, so we can solve for x:
(1/6)πx^3 = 4
πx^3 = 24
x^3 = 24/π
x = (24/π)^(1/3)
Now we can find the total volume of the ice cream:
Total volume = (1/4)πx^3 = (1/4)π((24/π)^(1/3))^3.
Using a calculator, we can evaluate this expression to find the total volume of the ice cream.