Use the image to answer the question.
A hemisphere is placed on top of a 3 D cylinder. The base radius is labeled r and the height of the cylinder is labeled h equals 2 r.
Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve this real-world problem:
A 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?
(1 point)
Responses
4y
4 y
2y
2 y
12y
12 y
8y
The volume of the whole building can be found by adding the volume of the cylinder and the volume of the hemisphere.
The volume of the cylinder is given as 6y.
The volume of a cylinder is calculated using the formula V = πr^2h, where r is the base radius and h is the height. In this case, we are given that the height of the cylinder is 2r.
Substituting the given values, the volume of the cylinder can be written as V = πr^2(2r) = 2πr^3.
The volume of a hemisphere is calculated using the formula V = 2/3πr^3.
Therefore, the volume of the whole building is 2πr^3 + 2/3πr^3 = (6/3 + 2/3)πr^3 = 8/3πr^3.
Since we are given that the volume of the cylinder is 6y, we can set up the following equation:
6y = 8/3πr^3
To find the volume of the whole building, we need to solve for y.
Dividing both sides of the equation by 6 gives:
y = (8/3πr^3) / 6 = 4/9πr^3.
Therefore, the volume of the whole building is 4/9πr^3.
In terms of y, the volume of the whole building is 4/9πr^3 = 4y.
Therefore, the correct answer is 4y.