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 5 sentences
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height. Since the diameter of the ice cream cone is equal to its height, we can let the height be h and the radius of the base be r. Then, the volume of the cone is V = (1/3)πr^2h.
The volume of a hemisphere is given by the formula V = (2/3)πr^3. In this case, the volume of the hemisphere is 4 in.^3, so we can solve the equation (2/3)πr^3 = 4 to find the radius of the hemisphere.
Once we have the radius of the hemisphere, we can use it as the radius of the base of the cone since the cone is placed on top of the hemisphere. Then, we can calculate the volume of the cone using the formula V = (1/3)πr^2h, where now r is the radius from the hemisphere and h is the same as the radius and the height of the cone.
Finally, we can add the volume of the cone to the volume of the hemisphere to get the total volume of ice cream in and on top of the cone.