Square pyramid water volume

Question

A square pyramid is filled with water to half it's height. Then it is reversed. What is the new height of water?

I found that the volume of the water is $7/8$ of the volume of the pyramid, but how do I find the new height in terms of the initial height?

Answer 1

If the pyramid has side $s$ and height $h$, the whole volume is $\frac 13s^2h$. By similar triangles, the pyramid the water occupies has side $as$ and height $ah$ for some value $a$. So $\frac 78 \cdot \frac 13s^2h=\frac 13 (as)^2(ah)$ Solve for $a$

Answer 2

We use a scaling argument. When the linear dimensions of an object are scaled by the scaling factor $\lambda$, areas are scaled by the factor $\lambda^2$, and volumes are scaled by the factor $\lambda^3$.

The new upside down pyramid of water is similar to the full pyramid, and the volume is scaled down by a factor of $7/8$. So linear dimensions are scaled down by a factor of $\sqrt[3]{7/8}$.