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.
Let's denote the radius of the cone and the hemisphere by R.
The volume of the cone is given by the formula: V_cone = (1/3) * π * R^2 * h, where 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 cone is equal to R/2.
The volume of the hemisphere is given by the formula: V_hemisphere = (2/3) * π * R^3.
We are given that the volume of the hemisphere is 4 in^3, so we can write the equation:
(2/3) * π * R^3 = 4
Simplifying the equation, we have:
(2/3) * π * R^3 = 4
(2/3) * π * R^3 = 3.14 * R^3 = 4
R^3 = 4 / 3.14
R^3 ≈ 1.273
Taking the cube root of both sides, we find:
R ≈ ∛1.273 ≈ 1.0775
Now, let's calculate the volume of the cone:
V_cone = (1/3) * π * R^2 * h
Since the diameter of the cone is equal to its height, we can say that h = R.
V_cone = (1/3) * π * (1.0775)^2 * 1.0775
V_cone ≈ 1.525 in^3
Finally, let's calculate the total volume of ice cream in and on top of the cone:
Total volume = V_cone + V_hemisphere
Total volume ≈ 1.525 in^3 + 4 in^3
Total volume ≈ 5.525 in^3
Therefore, the total volume of ice cream in and on top of the cone is approximately 5.525 in^3.