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. What is the exact volume of the frozen yogurt in the cone and hemisphere? Leave your answer in terms of pi. (1 point)
Responses
288π cm^3
360π cm^3
216π cm^3
144π cm^3
To find the volume of the frozen yogurt in the cone and hemisphere, we first need to find the volume of each shape separately and then add them together.
1. Volume of the cone:
The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height. Plugging in the given values, we have:
V_cone = (1/3)π(6^2)(24)
= (1/3)π(36)(24)
= 288π cm^3
2. Volume of the hemisphere:
The formula for the volume of a hemisphere is V = (2/3)πr^3. Since the radius of the hemisphere is the same as the radius of the cone, we have:
V_hemisphere = (2/3)π(6^3)
= (2/3)π(216)
= 144π cm^3
3. Volume of the frozen yogurt in the cone and hemisphere:
V_frozen yogurt = V_cone + V_hemisphere
= 288π + 144π
= 432π cm^3
Therefore, the exact volume of the frozen yogurt in the cone and hemisphere is 432π cm^3.