write this in 4 sentences
To solve this problem, we can use the relationship between the formulas for the volumes of cones and spheres.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height of the cone.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3)πr^3, where r is the radius of the hemisphere.
In this problem, the diameter of the ice cream cone is equal to its height. Let's denote the height (and diameter) of the cone as h and the radius of the cone as r.
From the given information, we know that the volume of the hemisphere is 4 in^3.
Using the formula for the volume of a hemisphere, we have 4 = (2/3)πr^3.
Solving for r, we get r^3 = (3/2)(4/π) = 6/π.
Taking the cube root of both sides, we find r = (6/π)^(1/3).
Since the diameter of the cone is equal to its height, we have r = h/2.
Substituting r = (6/π)^(1/3) into this equation, we get h/2 = (6/π)^(1/3).
Multiplying both sides by 2, we find h = 2 * (6/π)^(1/3).
Now that we know the height of the cone, we can calculate its volume using the formula for the volume of a cone.
V_cone = (1/3)πr^2h = (1/3)π((6/π)^(1/3))^2 * 2 * (6/π)^(1/3).
Simplifying this expression, we get V_cone = (2/3)π * 36/π^(2/3) * (6/π)^(1/3) = (2/3) * 36 * 6^(1/3) * π^(1/3).
Finally, to find the total volume of ice cream in and on top of the cone, we add the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere = (2/3) * 36 * 6^(1/3) * π^(1/3) + 4 = 24 * 6^(1/3) * π^(1/3) + 4.
So, the total volume of ice cream in and on top of the cone is 24 * 6^(1/3) * π^(1/3) + 4.