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 in 4-5 sentences.
To solve this problem, we can use the relationship between the volumes of cones and spheres. The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height. The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius.
Since the diameter of the ice cream cone is equal to its height, we can assume that the radius of the cone is equal to half the diameter. Let's say the radius of the cone is r1.
The volume of the cone can be calculated as V1 = (1/3)πr1^2h, where h is the height of the cone.
Now, the volume of the hemisphere on top of the cone is given as 4 in^3. Since a hemisphere is half of a sphere, we can calculate the radius of the hemisphere using the formula r2 = (3V2/4π)^(1/3), where V2 is the volume of the hemisphere.
The total volume of the ice cream in and on top of the cone can be calculated as V_total = V1 + V2.
Now we know the volume of the cone and the volume of the hemisphere, we can substitute the values and calculate the total volume of the ice cream.