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^2. 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.
(1 point)
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. The volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius.
Since the diameter of the cone is equal to its height, the radius of the cone would be half of the height. Let's denote the height (and diameter) of the cone as 2h, and the radius of the cone as h.
Therefore, the volume of the cone is V_cone = (1/3)πh^2(2h) = (2/3)πh^3.
Given that the volume of the hemisphere is 4 in^2, we can set up the equation:
(2/3)πh^3 = 4
Solving for h, we get h ≈ 1.396 inches.
Now, the total volume of ice cream in and on top of the cone would be the sum of the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere
Total volume = (2/3)π(1.396)^3 + (2/3)π(1.396)^3
Total volume ≈ 10.618 in^3
Therefore, the total volume of ice cream in and on top of the cone is approximately 10.618 in^3.