Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve this real-world problem:%0D%0A%0D%0AA video game developer is creating a building design composed of a cylinder and a hemisphere. She wants to know the volume of her design if the volume of the cylinder is 6y . What is the volume of the whole building?
To solve this problem, we need to find the volume of both the cylinder and the hemisphere separately, and then add them together to get the volume of the whole building.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius of the base of the cylinder, and h is the height of the cylinder.
The formula for the volume of a hemisphere is V_hemisphere = (2/3)πr^3, where r is the radius of the hemisphere.
Let's assume the radius of both the cylinder and the hemisphere is r.
Given that the volume of the cylinder is 6y, we can equate it to the volume formula:
6y = πr^2h
Similarly, we need to find the volume of the hemisphere. Since the hemisphere is half of a sphere, we can use the formula for the volume of a sphere, which is 4/3 πr^3, and divide it by 2 to get the volume of the hemisphere:
V_hemisphere = (1/2) * (4/3)πr^3
V_hemisphere = (2/3)πr^3
Now, to find the volume of the whole building, we need to add the volume of the cylinder and the hemisphere:
Volume of the building = Volume of the cylinder + Volume of the hemisphere
Volume of the building = 6y + (2/3)πr^3
So, the volume of the whole building is 6y + (2/3)πr^3.