Question $1(50)$ marks$)$

The demand for electric power is usually much higher during the day than it is at night, and

utility companies often sell power at night at much lower prices to encourage consumers

to use the available power generation capacity and to avoid building new, expensive power

plants that will be used only a short time during peak periods. Utilities are also willing to

purchase power produced during the day from private parties at a high price.

Suppose a utility company is selling electrical power for $$\frac{0\mathrm{.}03}{k}Wh$ at night and is willing

to pay $$\frac{0\mathrm{.}08}{k}Wh$ for power produced during the day. To take advantage of this

opportunity, an entrepreneur is considering building a large reservoir $40m$ above the lake

level, pumping water from the lake to the reservoir at night using cheap power, and letting

the water flow from the reservoir back to the lake during the day, producing power as the

pump$-$motor operates as a turbine$-$generator during reverse flow. Preliminary analysis

shows that a water flow rate of $2\frac{m^3}{s}$ can be used in either direction, and the irreversible

headloss of the piping system is $4m.$ Assume the efficiency of the pump$-$motor system is

$85\%,$ the efficiency of the turbine$-$generator system is $75\%$ and that the system operates

for $10h$ in pump mode and $10h$ in turbine mode each day.

Will this system generate revenue, or operate at a loss? Determine the annual profit

$($loss$).$