The frozen yogurt cones 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 and terms of pi
To find the volume of the frozen yogurt in the cone and hemisphere, we need to find the volumes of the cone and 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,
where r is the radius of the cone and h is the height of the cone.
Substituting the given values:
V_cone = (1/3) * π * (6^2) * 24
= (1/3) * π * 36 * 24
= 288π cm^3
The volume of a hemisphere is given by the formula:
V_hemisphere = (2/3) * π * r^3,
where r is the radius of the hemisphere.
Substituting the given value:
V_hemisphere = (2/3) * π * (6^3)
= (2/3) * π * 216
= 144π cm^3
Now, adding the volumes together:
V_total = 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.