## Precalculus (6th Edition) Blitzer

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.