Let's denote the height and diameter of the ice cream cone as h and d, respectively.
The volume of a cone is given by the formula: V(cone) = (1/3) * π * r^2 * h, where r is the radius of the cone.
Since the diameter of the cone is equal to its height, we have d = h. Therefore, the radius is half the diameter, so r = d/2.
The volume of a sphere is given by the formula: V(sphere) = (4/3) * π * r^3.
We are given that the hemisphere on top of the cone has a volume of 4 in^3. This is half the volume of a complete sphere, so we can write:
4 = (1/2) * [(4/3) * π * (d/2)^3]
Simplifying this equation:
4 = (1/2) * [(4/3) * π * (d^3/8)]
Multiplying through by 2/Ï€:
8/Ï€ = (4/3) * (d^3/8)
Multiplying through by 3/4:
6/Ï€ = d^3/8
Multiplying through by 8:
48/Ï€ = d^3
Taking the cube root of both sides:
d = (48/Ï€)^(1/3)
Now, we can substitute the value of d back into the formula for the volume of the cone:
V(cone) = (1/3) * π * (d/2)^2 * d
Simplifying this equation:
V(cone) = (1/3) * π * (d^2/4) * d
V(cone) = (Ï€/12) * d^3
Now, we can find the total volume of the ice cream in and on top of the cone by adding the volume of the cone and the volume of the hemisphere:
Total volume = V(cone) + V(hemisphere)
Total volume = (π/12) * d^3 + (1/2) * [(4/3) * π * (d/2)^3]
Total volume = (π/12) * d^3 + (2/3) * π * (d/2)^3
Total volume = (π/12) * d^3 + (1/6) * π * d^3
Total volume = (7/12) * π * d^3
Total volume = (7/12) * π * [(48/π)^(1/3)]^3
Total volume = (7/12) * π * (48/π)
Total volume = (7/12) * 48
Total volume = 7 * 4
Total volume = 28 in^3
Therefore, the total volume of ice cream in and on top of the cone is 28 in^3.