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?
The volume of a solid cone is given by the formula (1/3)πr^2h, where:
- π is a constant approximately equal to 3.14159
- r is the radius of the circular base of the cone
- h is the height of the cone
Let's assume that the radius of the cone is r and the height of the cone is h.
Since the volume of the cone is 5x, we have:
(1/3)πr^2h = 5x
Multiplying both sides of the equation by 3/π, we get:
r^2h = (15/π)x
The height of the cone is h, and the radius of the circular base of the cone is r, which is also the radius of the cylinder and the hemisphere.
The volume of the cylinder is given by the formula πr^2h, so the volume of the cylinder is x.
The volume of the hemisphere is given by the formula (2/3)πr^3, so the volume of the hemisphere is (2/3)πr^3.
Therefore, the total volume of the toy model is:
(1/3)πr^2h + πr^2h + (2/3)πr^3
Combining like terms:
(1/3)πr^2h + (3/3)πr^2h + (2/3)πr^3
Simplifying the fractions:
(1 + 3 + 2/3)πr^2h
(10/3)πr^2h
Since the volume of the cone is 5x, we can substitute (15/π)x for r^2h:
(10/3)π(15/π)x
(150/3)x
50x
Therefore, the total volume of the toy model is 50x.