How do you find the volume of a right circular cone with a hemisphere? The height of the cone is 8 cm and the radius in 4cm.

To find the volume of a composite shape such as a right circular cone with a hemisphere, you can break it down into separate components and then add their volumes.

Step 1: Find the volume of the cone
The volume of a cone can be calculated using the formula:
Volume_of_cone = 1/3 * π * (radius of the base)^2 * height

Given:
Radius of the base (cone) = 4 cm
Height of the cone = 8 cm

Using the formula, we can calculate the volume of the cone:
Volume_of_cone = 1/3 * π * (4 cm)^2 * 8 cm
= 1/3 * π * 16 cm^2 * 8 cm
= 1/3 * π * 128 cm^3
= 128/3 * π cm^3 (approximately 134.04 cm^3)

Step 2: Find the volume of the hemisphere
The volume of a hemisphere can be calculated using the formula:
Volume_of_hemisphere = 2/3 * π * (radius of the hemisphere)^3

Given:
Radius of the hemisphere = 4 cm

Using the formula, we can calculate the volume of the hemisphere:
Volume_of_hemisphere = 2/3 * π * (4 cm)^3
= 2/3 * π * 64 cm^3
= 128/3 * π cm^3 (approximately 134.04 cm^3)

Step 3: Add the volumes of the cone and hemisphere
To find the total volume of the right circular cone with a hemisphere, we add the volumes of the cone and hemisphere:
Total_volume = Volume_of_cone + Volume_of_hemisphere
= 128/3 * π cm^3 + 128/3 * π cm^3
= (128/3 + 128/3) * π cm^3
= (256/3) * π cm^3 (approximately 267.87 cm^3)

Therefore, the volume of the right circular cone with a hemisphere is approximately 267.87 cm^3.