Posted by Mick on Monday, February 14, 2011 at 7:07pm.
Why do you use 1/3 in volume formulas?
geometry - Reiny, Monday, February 14, 2011 at 7:34pm
any solid that comes to a vertex or point has the formula
V = (1/3) (base area) x (height)
I you know Calculus you could take the region bounded by the x-axis, the y-axis and a line with x-intercept of h and y-intercept of r and rotate it about the x-axis
The result will be a cone with radius r and height h
the equation of that line is
y = (-r/h)x + r
the generated volume
= π[integral]y^2 dx
= π[integral] (r^2x^2/h^2 - 2r^2x/h + r^2) dx
= π [r^2x^3/(3h^2) - r^2x^2/h + r^2x] from 0 to h
= π(r^2h/3 - r^2h + r^2h - 0)
= (1/3)(πr^2)(h) or (1/3)basearea x height
When I was still teaching, I had a cylinder and a cone, both with the same radius and height.
In class we would fill the cone with water and pour it into the cylinder, and do that until the cylinder was full.
We were able to fill and pour THREE times, showing that the volume of the cone was
(1/3) of the volume of the cylinder.
volume of cylinder = πr^2h
volume of cone = (1/3)πr^2h
Answer This Question
More Related Questions
- geometry - A hemispherical dome holds a volume of 486π..find the surface ...
- geometry - Looking at formulas, which 2 shapes describe this relationship: the ...
- Intro to computers - You are an inexperienced Excel user and don’t know how to ...
- Geo - Hi I need some major help on surface area and volume. Do you think you can...
- computer applications - John wants to see the formulas he has used in his ...
- Math - When something says that I'll be tested on formulas for perimeter and ...
- Statistics - Hi, My professor gave us the formulas for sample variance and ...
- math - Okay. So i'm doing this project for my geometry class about different ...
- math - Can you describe some common formulas that you use in your life? Like ...
- Math - Use the formulas V=s❸ andA=6s❷ to calculate the volume and ...