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. S
To find the total volume of ice cream in and on top of the cone, we need to find the volume of the cone and the volume of the hemisphere separately and then add them together.
The volume of a cone is given by the formula V_cone = (1/3) * pi * r^2 * h, where r is the radius and h is the height.
From the image, we can see that the diameter of the ice cream cone is equal to its height, so we can let d be the diameter and h be the height.
Since the radius is half the diameter, we have r = (1/2)d = (1/2)h.
The volume of the cone is then V_cone = (1/3) * pi * [(1/2)h]^2 * h = (1/12) * pi * h^3.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3) * pi * r^3, where r is the radius.
Since the radius of the hemisphere is the same as the radius of the cone, we can substitute (1/2)h for r in the volume formula for the hemisphere.
The volume of the hemisphere is then V_hemisphere = (2/3) * pi * [(1/2)h]^3 = (1/24) * pi * h^3.
Adding the volume of the cone and the volume of the hemisphere, we get the total volume of ice cream in and on top of the cone: V_total = V_cone + V_hemisphere = (1/12) * pi * h^3 + (1/24) * pi * h^3 = (1/8) * pi * h^3.
Given that the volume of the hemisphere is 4 in^3, we can set up the equation (1/24) * pi * h^3 = 4 and solve for h.
Multiplying both sides by 24/π, we get (1/8) * h^3 = 4 * 24/π.
Simplifying, we have h^3 = 32 * 24/π.
Taking the cube root of both sides, we have h = (32 * 24/π)^(1/3).
Substituting this value of h back into the equation for the total volume of ice cream, we get V_total = (1/8) * pi * [(32 * 24/π)^(1/3)]^3.
Simplifying further, we have V_total = (1/8) * pi * (32 * 24/π) = 384/8 = 48 in^3.
Therefore, the total volume of ice cream in and on top of the cone is 48 in^3.