Geometry and measurement question. A square pyramid tank (base width 6 m; height 3 m), sitting on its square base, has 1 m depth of water. Suppose the tank is inverted and is standing on its apex. What is the new depth of water.
To me it looks like, it will have same same depth, since you are just inverting it and there is no size change in cylinder measurements.
Please help.
 A: Unfortunately, my cat ate my formula sheet, and I don't remember the formula for the area of a pyramid. But let's see what can be done.
The big pyramid has height $3$.  The empty space above the water is a pyramid similar to the big one, but with height $2$. 
Let $V$ be the volume of the big pyramid. Then the empty space has volume $\dfrac{8}{27}V$. For if linear dimensions are scaled by the factor $\rho$, then volumes are scaled by the factor $\rho^3$. Thus the volume of water is $\dfrac{19}{27}V$.
Invert the pyramid, and let $h$ be the height of the pyramid of water. The ratio of the linear dimensions of the water pyramid to the big pyramid is $\dfrac{h}{3}$. So the volume of the water pyramid is $\dfrac{h^3}{27}V.$
It follows that 
$$\frac{19}{27}V=\frac{h^3}{27}V.$$
Thus $h=\sqrt[3]{19}$.
Remarks: $1$. This is a nice example of a scaling argument.  It can be further simplified, by calling the volume of the original pyramid $27$. What is the unit of volume?  Maybe we can call it a pharaoh. 
$2$. Suppose that our container is a generalized cone of height $3$. A  generalized cone is obtained by first drawing a closed curve $B$ on the ground (in the case of our pyramid, a square). Then take a point $P$ which is $3$ metres above ground level, and join $P$ to all the points of $B$.  Exactly the same argument shows that if our container is this kind of generalized cone, and the original height of water is $1$ metre, then when we invert the height of the water is $\sqrt[3]{19}$.
$3.$ If you happen to know the formula for the volume of a pyramid, you can use it to find the answer. After some unnecessarily complicated calculation, you will get the answer $\sqrt[3]{19}$. 
