The frozen yogurt cone shown is made from a cone and a hemisphere. The radius is 6 cm. The total height of the cone is 24 cm. The cone is completely packed with frozen yogurt.
To find the volume of the frozen yogurt cone, we need to find the volume of the cone and the volume of the hemisphere and then add them together.
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.
Given that the radius of the cone is 6 cm and the total height of the cone is 24 cm, we can substitute these values into the formula:
V_cone = (1/3)π(6^2)(24)
= (1/3)π(36)(24)
= (1/3)(36π)(24)
= 288π cm^3
The volume of a hemisphere is given by the formula:
V_hemisphere = (2/3)πr^3
Given that the radius of the hemisphere is 6 cm, we can substitute this value into the formula:
V_hemisphere = (2/3)π(6^3)
= (2/3)π(216)
= (2/3)(216π)
= 144π cm^3
To find the total volume of the frozen yogurt cone, we add the volume of the cone and the volume of the hemisphere:
V_total = V_cone + V_hemisphere
= 288π + 144π
= 432π cm^3
Therefore, the frozen yogurt cone is completely packed with 432π cm^3 of frozen yogurt.