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
To find the total volume of the toy model, we need to add the volumes of the cone, cylinder, and hemisphere.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the base radius and h is the height.
We are given that the volume of the cone is 5x, so we can write:
5x = (1/3)πr^2(2r)
Simplifying this equation, we have:
15x = πr^3
Now, the volume of a cylinder is given by the formula V = πr^2h, where r is the base radius and h is the height.
We are given that the height of the cylinder is 2r, so we can write:
V_cylinder = πr^2(2r) = 2πr^3
Finally, the volume of a hemisphere is given by the formula V = (2/3)πr^3, where r is the radius.
We have a hemisphere with the same radius as the cylinder, so we can write:
V_hemisphere = (2/3)πr^3
To find the total volume of the toy model, we add the volumes of the cone, cylinder, and hemisphere:
Total volume = V_cylinder + V_cone + V_hemisphere
= 2πr^3 + 5x + (2/3)πr^3
Combining like terms, we have:
Total volume = (2π/3)r^3 + 5x + (2/3)πr^3
To simplify further, we can factor out (2/3)πr^3 and combine it with the other term:
Total volume = [(2π/3)r^3 + (2/3)πr^3] + 5x
= (4/3)πr^3 + 5x
Therefore, the total volume of the toy model is (4/3)πr^3 + 5x.
None of the given options match this expression, so none of the provided answers are correct.