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.
(4 points)
in 4 sentences
To find the total volume of ice cream, we need to find the volume of the cone and the hemisphere separately and then add them together. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. Since the diameter of the cone is equal to its height, the radius is half the diameter.
So, the volume of the cone is (1/3)π(d/2)^2(d/2) = (1/3)π(d/2)^3.
The volume of the hemisphere is given by the formula V = (2/3)πr^3. Since the diameter of the hemisphere is the same as the diameter of the cone, the radius of the hemisphere is also half the diameter.
So, the volume of the hemisphere is (2/3)π(d/2)^3.
Adding these two volumes together, we get the total volume of ice cream in and on top of the cone.