#### Answer

2 hours.

#### Work Step by Step

The number of hours you sleep each night varies inversely as the square of the number of cups of coffee consumed during the early evening.
Let t be the number of hours you sleep and n be the number of cups of coffee consumed.
Then, we have
$\begin{align}
& t\propto \frac{1}{{{n}^{2}}} \\
& \text{or,}\,\,t=\frac{k}{{{n}^{2}}}\,\,\,\,\,\,\left[ \text{where}\,\,\text{k}\,\,\text{is}\,\,\text{any}\,\,\text{constant} \right] \\
& \text{for,}\,\,n=2\,\,\,and\,\,t=\,8,\text{ we can have,} \\
& \text{or,}\,\,8=\frac{k}{{{2}^{2}}}
\end{align}$
And,
$\begin{align}
& \text{or,}\,\,k=8\times 4 \\
& \text{or,}\,\,k=32 \\
\end{align}$
Now, the number of cups of coffee is doubled, i.e. n = 4. So,
$\begin{align}
& t=\frac{k}{{{\left( 4 \right)}^{2}}} \\
& \text{or,}\,\,\,t\,=\frac{32}{16}\,\,\,\,\,\,\,\left[ k=32 \right] \\
& \text{or,}\,\,\,t=2
\end{align}$
Thus, you can sleep 2 hours.