A 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
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_cone
where r is the radius of the cone and h_cone 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 plug in these values into the formula to find the volume of the cone:
V_cone = (1/3) * π * 6^2 * 24
V_cone = 1/3 * π * 36 * 24
V_cone = 288 * π 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.
Given that the radius of the hemisphere is also 6 cm, we can plug in this value into the formula to find the volume of the hemisphere:
V_hemisphere = (2/3) * π * 6^3
V_hemisphere = 2/3 * π * 216
V_hemisphere = 144 * π cm^3
Now, we can add the volume of the cone and the volume of the hemisphere to find the total volume of the frozen yogurt:
V_total = V_cone + V_hemisphere
V_total = 288 * π + 144 * π
V_total = 432 * π cm^3
Therefore, the exact volume of the frozen yogurt in the cone and hemisphere is 432 * π cm^3.