Cones, Cylinders, & Spheres Unit Test
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Question
Use the image to answer the question.
A hemisphere is placed on the left of a horizontally placed 3 D cylinder. A cone is placed on the right side of the cylinder. The base radius of the cylinder is labeled r, the height of the cylinder between the two bases is labeled h equals 2 r, and the height of the cone from the base to the apex is labeled h equals 2 r.
A toy designer is creating a 3D model of a pencil composed of a cone, cylinder, and hemisphere. She knows that the volume of the cone is 5x . What is the total volume of the toy model?
(1 point)
Responses
30x
30 x
20x
20 x
15x
15 x
25x
25 x
To find the total volume of the toy model, we need to find the volumes of the cone, cylinder, and hemisphere and then add them together.
The volume of a cone is given by the formula V = 1/3πr^2h, where r is the radius of the base and h is the height. In this case, the given volume of the cone is 5x, so we can set up the equation:
5x = 1/3πr^2h
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the base and h is the height. In this case, the height of the cylinder is 2r, so we can substitute that into the formula:
V = πr^2(2r)
V = 2πr^3
The volume of a hemisphere is given by the formula V = 2/3πr^3, where r is the radius of the hemisphere. In this case, the radius is the same as the radius of the cylinder, so we can substitute that into the formula:
V = 2/3πr^3
Now we can add the volumes together:
Total volume = 5x + 2πr^3 + 2/3πr^3
We cannot simplify this any further without knowing the value of r.