Use the image to answer the question.
A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
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.
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_cone = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3)πr^3, where r is the radius of the hemisphere.
In this problem, the diameter of the ice cream cone is equal to its height. Let's denote the height (and diameter) of the cone as h and the radius of the cone as r.
From the given information, we know that the volume of the hemisphere is 4 in^3.
Using the formula for the volume of a hemisphere, we have 4 = (2/3)πr^3.
Solving for r, we get r^3 = (3/2)(4/π) = 6/π.
Taking the cube root of both sides, we find r = (6/π)^(1/3).
Since the diameter of the cone is equal to its height, we have r = h/2.
Substituting r = (6/π)^(1/3) into this equation, we get h/2 = (6/π)^(1/3).
Multiplying both sides by 2, we find h = 2 * (6/π)^(1/3).
Now that we know the height of the cone, we can calculate its volume using the formula for the volume of a cone.
V_cone = (1/3)πr^2h = (1/3)π((6/π)^(1/3))^2 * 2 * (6/π)^(1/3).
Simplifying this expression, we get V_cone = (2/3)π * 36/π^(2/3) * (6/π)^(1/3) = (2/3) * 36 * 6^(1/3) * π^(1/3).
Finally, to find the total volume of ice cream in and on top of the cone, we add the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere = (2/3) * 36 * 6^(1/3) * π^(1/3) + 4 = 24 * 6^(1/3) * π^(1/3) + 4.
So, the total volume of ice cream in and on top of the cone is 24 * 6^(1/3) * π^(1/3) + 4.