An off$-$grid, pumped hydro system, powered using a photovoltaic array, is designed to provide $\backslash (10\backslash$mathrm$\{$kWh$\}/\backslash )$ day for a small cabin located near a pond, with a peak demand of $3$ kW $.$ A storage tank is placed on a nearby hilltop $100$ feet above the turbine$/$generator$.$ Pipe connections, each $800$ feet in length, run to and from the tank.

a$.$ Assuming an average head loss of $\backslash (15\backslash \%\backslash )$ in the pipeline, and $\backslash (80\backslash \%\backslash )$ conversion efficiency for the turbine$/$generator$,$ how many gallons of water need to flow from the upper tank to the pond to supply the daily energy requirement for the cabin?

b$.$ If the piping head losses are $20\backslash \%$ during peak demand, what flow rate of water $($in gpm$,$ or gallons per minute$)$ from the tank is needed to supply the $3$ kW peak?

c$.$ What size PVC pipe $($in$)$ will keep the friction head losses to $\backslash (20\backslash \%\backslash )$ at peak demand? $($Use the data provided in Figure $8\mathrm{.}35$ of Masters $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ edition.$)$

d$.$ The photovoltaic array uses modules comprised of gallium arsenide $($GaAs$)$ cells, which have a band$-$gap energy of $1\mathrm{.}43$ eV $.$

How much $($eV$)$ of the energy from a photon with an energy of $1\mathrm{.}12$ eV can be used to excite an electron to the GaAs conduction band?

How much $($eV$)$ of the energy from a photon with an energy of $1\mathrm{.}67$ eV can be used to excite an electron to the GaAs conduction band?

Give a brief explanation of your answers.