The diagram shows a pyramid with a rectangular base and its height is 9 cm. The volume of the liquid in the pyramid is $\tfrac{2}{3}$ the volume of the pyramid. Calculate the depth of the liquid.

1. Find the volume ratio of the empty smaller pyramid to the whole pyramid.

Since the liquid is $\dfrac{2}{3}$ of the volume of the whole pyramid,

$\text{volume of empty top pyramid}=1-\dfrac{2}{3}=\dfrac{1}{3}$

of the volume of the whole pyramid.

2. Use volume ratio of similar pyramids to find the height of the empty top pyramid.

Let the height of the empty top pyramid be $h$ cm.

The empty top pyramid is similar to the whole pyramid, so

$\left(\frac{h}{9}\right)^3=\frac{1}{3}$

$\frac{h}{9}=\sqrt[3]{\frac{1}{3}}$

$h=9\sqrt[3]{\frac{1}{3}}$

3. Calculate the depth of the liquid from the base.

The depth of the liquid is

$9-h=9-9\sqrt[3]{\frac{1}{3}}$

$=2.759\ldots$

$\approx 2.76\text{ cm}$