Aimee packs ice cream into an ice cream cone. Then she puts a perfect hemisphere of ice cream on top of the cone that has the 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 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
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 and h is the height of the cone.
Since the diameter of the cone is equal to its height, we can say that the radius of the base of the cone is equal to half of its height. Let's denote the radius as r and the height as 2r.
The volume of the cone can now be written as V = (1/3)πr^2(2r) = (2/3)πr^3.
We are given that the volume of the hemisphere on top of the cone is 4 in.^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 radius of the sphere is the same as the radius of the cone (because it is a perfect hemisphere), we can set the volume of the hemisphere equal to the volume of the cone and solve for r.
(2/3)πr^3 = 4.
Dividing both sides by (2/3)π gives r^3 = (3/2)(4/π) = 6/π.
Taking the cube root of both sides gives r = (6/π)^(1/3).
Now that we have the radius of the cone (and the hemisphere), we can calculate the volume of the cone and hemisphere together.
The volume of the cone is given by V_cone = (2/3)πr^3 = (2/3)π(6/π)^(1/3)^3 = (2/3)π(6/π) = 4 in.^3.
The volume of the hemisphere is 4 in.^3.
Therefore, the total volume of ice cream on top of the cone is 4 + 4 = 8 in.^3.
Can you shorten your explanation?
Sure! Here is a shorter explanation:
The volume of a cone is (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 denote the radius as r and the height as 2r. The volume of the cone is then (2/3)πr^3.
We are given that the volume of the hemisphere on top of the cone is 4 in.^3, which is also the volume of the cone. Setting the volumes equal to each other and solving for r gives r = (6/π)^(1/3).
The volume of the cone is then (2/3)π(6/π)^(1/3)^3 = 4 in.^3.
The volume of the hemisphere is 4 in.^3.
Therefore, the total volume of ice cream on top of the cone is 4 + 4 = 8 in.^3.