# Thread: fun volume problem

1. ## fun volume problem

Here is a cool volume problem if anyone would like a go.

A cylindrical barrel of diameter d lies on a stand so that its axis makes an angle of 20 degrees with the horizontal. The amount of fluid in the barrel just covers the bottom of the barrel. Approximate the volume of fluid in the barrel

2. Hello, galactus!

A cylindrical barrel of diameter $d$ lies on a stand
so that its axis makes an angle of 20° with the horizontal.
The amount of fluid in the barrel just covers the bottom of the barrel.
Approximate the volume of fluid in the barrel.
Code:
                                          D
*
*      *
L      *             * d
*                    *
*                           *
*                                  *
A* - - - - - - - - - - - - - - - - - - - - *C
:*::::::::::::::::::::::::::::::::::*     :
: *:::::::::::::::::::::::::::*           :
:  *d:::::::::::::::::::* L               :
:20°*:::::::::::::*                       :
:    *::::::*  20°                        :
P+ - - * - - - - - - - - - - - - - - - - - +Q
B
Let $L$ = length of the cylinder.

In $\Delta APB\!:\;\;AP \:=\:d\cos20^o$

In $\Delta CQB\!:\;\;CQ \:=\:L\sin20^O$

Since $CQ = AP,\;\;L\sin20^o \:=\:d\cos20^o \quad\Rightarrow\quad L \;=\;d\cot20^o$

The fluid occupies half the volume of the cylinder: . $V \;=\;\frac{1}{2}\cdot\pi\left(\frac{d}{2}\right)^2 L$

Therefore: . $V \;=\;\frac{1}{2}\cdot\pi\cdot\frac{d^2}{4}(d\cot20 ^o) \;=\;\frac{\pi}{8}d^3\cot20^o$

3. Didn't take you long, Soroban. Good man. That's what I got. It wasn't particularly difficult, just a cool problem.

4. This problem is relatively easy using geometry and trig. Can you do it by integrating (using calculus)?