Final answer:
The exact total volume of the frozen yogurt in both the cone and the hemisphere is 432π cm³ when the radius is 6 cm and the total height of the cone is 24 cm.
Step-by-step explanation:
To find the exact volume of the frozen yogurt in both the cone and the hemisphere, we will use the formulas for the volume of a cone and the volume of a hemisphere.
Volume of a Cone
The volume of a cone is calculated using the formula:
Vcone = (1/3)πr²h
Where r is the radius and h is the height of the cone.
Given the radius (r) is 6 cm and the total height of the cone (h) is 24 cm, the volume of the cone is:
Vcone = (1/3)π(6 cm)²(24 cm)
Vcone = 288π cm³
Volume of a Hemisphere
The volume of a hemisphere is half the volume of a sphere, calculated using the formula:
Vhemisphere = (1/2)(4/3)πr³
Using the given radius (r) of 6 cm:
Vhemisphere = (1/2)(4/3)π(6 cm)³
Vhemisphere = 144π cm³
The exact total volume of the frozen yogurt is the sum of the volumes of the cone and the hemisphere:
Total Volume = Vcone + Vhemisphere = 288π cm³ + 144π cm³
Total Volume = 432π cm³