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.%0D%0A%0D%0A%0D%0A%0D%0AWhat is the exact volume of the frozen yogurt in the cone and hemisphere? Leave your answer in terms of pi.
To find the volume of the frozen yogurt cone, we first need to find the volume of the cone and the hemisphere separately, and then add them together.
The volume of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h_cone
where r is the radius of the cone and h_cone is the height of the cone.
Substituting the given values, we have:
V_cone = (1/3) * π * (6)^2 * 24
= (1/3) * π * 36 * 24
= (1/3) * π * 864
= 288π
The volume of a hemisphere is given by the formula:
V_hemisphere = (2/3) * π * r^3
Substituting the given radius, we have:
V_hemisphere = (2/3) * π * (6)^3
= (2/3) * π * 216
= 144π
Therefore, the volume of the frozen yogurt in the cone and hemisphere is:
V_total = V_cone + V_hemisphere
= 288π + 144π
= 432π
So the exact volume of the frozen yogurt in the cone and hemisphere is 432π cubic cm.