f the cone below is filled with 4 in cubed of ice cream and we add a hemisphere of ice cream on top, how much is the total volume of ice cream? (The diameter of the cone is equal to its height.)
3 points
8 inches
8 in cubed
4 in cubed
To find the total volume of ice cream, we need to calculate the volume of the cone and the volume of the hemisphere separately.
For the cone, we can use the formula for the volume of a cone: V_cone = (1/3) * π * r^2 * h
Since the diameter of the cone is equal to its height, the radius (r) is half of the diameter, so r = 8/2 = 4 inches. The height (h) is also 8 inches. Now we can calculate the volume of the cone:
V_cone = (1/3) * π * 4^2 * 8
V_cone = (1/3) * π * 16 * 8
V_cone = (1/3) * 128π
V_cone = 128π/3 cubic inches
For the hemisphere, we can use the formula for the volume of a hemisphere: V_hemisphere = (2/3) * π * r^3
Again, the radius (r) is half of the diameter, so r = 4 inches. Now we can calculate the volume of the hemisphere:
V_hemisphere = (2/3) * π * 4^3
V_hemisphere = (2/3) * π * 64
V_hemisphere = (2/3) * 64π
V_hemisphere = 128π/3 cubic inches
Finally, we can find the total volume by adding the volume of the cone and the volume of the hemisphere:
Total Volume = V_cone + V_hemisphere
Total Volume = 128π/3 + 128π/3
Total Volume = 256π/3 cubic inches
So, the total volume of ice cream is 256π/3 cubic inches, or approximately 268.09 cubic inches.