The volume of a cone is given by the formula V = (1/3)Ï€r^2h, where r is the radius and h is the height. The volume of a hemisphere is given by the formula V = (2/3)Ï€r^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)Ï€h^2(2h) = (2/3)Ï€h^3.
Given that the volume of the hemisphere is 4 in^2, we can set up the equation:
(2/3)Ï€h^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)Ï€(1.396)^3 + (2/3)Ï€(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.