$2.$ A utility company sells power for $\backslash ($ r $\backslash )$ sell $\backslash (=3\backslash )$ cents per kilowatt$-$hour at night and is willing to buy power at a higher rate from independent energy suppliers during the day. An entrepreneur is considering building a large reservoir a distance $\backslash ($ H $\backslash )$ above a lake. He will pump water from the lake to the reservoir at night. During the day he will let the water flow from the reservoir to the lake, producing power by running the pump in reverse so that it operates as a turbine. The system operates in pump and turbine modes for the same length of time, $\backslash ($ T $\backslash ),$ the mass flow rate, $\backslash (\backslash$dot$\{\backslash$mathrm$\{$m$\}\}\backslash ),$ is the same for pump and turbine modes and the head loss is $\backslash ($ h $\backslash$mathrm$\{$~L$\}=1/3$ H $\backslash )$ for pump and turbine modes. For a quick estimate, you can assume that the reservoir depth is small compared to $\backslash ($ H $\backslash ),$ the reservoir fills and empties very slowly, and the lake depth is constant. If the power utility buys power for $\backslash ($ r b u y$=7\backslash )$ cents per kilowatt$-$hour, will the entrepreneur's company be profitable? HINT: The costs of pump and turbine modes are $\backslash ($ W $\backslash$mathrm$\{$p$\}$ T r $\backslash )$ sell and $\backslash ($ W $\backslash$mathrm$\{$tTrbuy$\}\backslash ),$ respectively.