# Which formula you reckon?!

• May 24th 2008, 05:06 PM
User Name
Which formula you reckon?!
Hi guys which formula suit this Question?
thanks(Itwasntme)
• May 24th 2008, 06:26 PM
Soroban
Hello, User Name!

I'll number the formulas like this: . $\displaystyle \begin{array}{cc} (1) & (2) \\ (3) & (4) \\ (5) \end{array}$

Formula (1): .$\displaystyle V \;=\;\pi\left(\frac{a+b}{2}\right)^2$
. . $\displaystyle \pi r^2$ is the area of the circle. .$\displaystyle \frac{a+b}{2}$ is the average of the two radii.

Therefore, (1) gives the area of the "average circle".
. . I seriously doubt that this happens to be the volume of the doughnut.

Formula (3): .$\displaystyle V \:=\:\frac{\pi}{3}(a^3 + b^3)$
. . $\displaystyle \frac{4}{3}\pi r^3$ is the volume of a sphere.

. . We have: .$\displaystyle \frac{1}{4}\left[\frac{4}{3}\pi a^3 + \frac{4}{3}\pi b^3\right]$

This is the volume of a sphere of radius $\displaystyle a$
. . plus the volume of a sphere of radius $\displaystyle b$
. . divided by 4.

This is too large to be the volume of the doughnut.

Formula (4): .$\displaystyle V \;=\;\pi^3(b^2-a^2)$

We have: .$\displaystyle \pi^2\left(\pi b^2 - \pi a^2\right)$

. . $\displaystyle \pi b^2$ is the area of the outer circle.
. . $\displaystyle \pi a^2$ is the area of the inner circle.
. . $\displaystyle \pi b^2 - \pi a^2$ is the area of the "ring".

I don't think multiplying by $\displaystyle \pi^2$ will give us the volume of the dougnut.

Formula (5): .$\displaystyle V \;=\;\frac{\pi^2}{3}(a + b)^3$
We have: .$\displaystyle \frac{\pi}{4}\left[\frac{4}{3}\pi (a+b)^3\right]$

This the volume of a sphere of radius $\displaystyle a+b$ ... (quite large!)
. . multipled by $\displaystyle \frac{\pi}{4}$

This is far too large to be the volume of the doughtnut.

By elimination, .$\displaystyle (2)\;\;V \:=\:\frac{\pi^2}{4}(a+b)(b-a)^2$ .is the volume of the doughnut.

• May 24th 2008, 06:39 PM
User Name