## 54. Math Puzzle 11/69

Two cans, filled with coffee, have the same shape and are made of the same material. The first can has a mass of $2 \mathrm{~kg}$ and is $12.0 \mathrm{~cm}$ high; the second has a mass of $1 \mathrm{~kg}$ and is $9.5 \mathrm{~cm}$ high.

How much coffee do the cans contain?
Let the mass of the content of the first can be $x \mathrm{~kg}$, that of the smaller one $y \mathrm{~kg}$. The masses of the cans we denote with $z$ and $t$ kg, respectively. Thus, we obtain the equations: $x+z=2$ and $y+t=1$.

Since the masses of the contents of the full containers are proportional to their volumes, we have $\frac{x}{y}=\frac{12^{3}}{9.5^{3}} \approx$ 2.02 or $x=2.02 y$.

The masses of the empty cans, however, are proportional to their surface areas. Therefore, $\frac{z}{t}=$ $\frac{12^{2}}{9.5^{2}} \approx 1.60$ or $t=1.60 t$.

By substituting $x$ and $z$ into the first equation, we obtain the system of equations

$$
\begin{array}{r}
2.02 y+1.60 t=2 \\
y+t=1
\end{array}
$$

with the solution $y=0.95, t=0.05$. By substituting into the respective equations, we get $x=1.92, z=0.08$. In the larger can, there are $1.92 \mathrm{~kg}$ and in the smaller one $0.94 \mathrm{~kg}$ of coffee contained.