Use the deductive problem solving strategy to solve the following problems:

$(25$ points$)$ Determine the how much more you are willing to spend for the purchase of the optimum gasoline engine with a thermal efficiency of $40\%.$

The budget engine has a thermal efficiency of $30\%.$ Fuel is expected to cost $$5$ per gallon over the $15$ year life of the engine. The density of gasoline is $750[$kg$/$m$3]$ with a lower heating value of $45,000[$kJ$/$kg$].$ The engine operates $1000[$hrs$/$yr$]$ at an average power output of $10[$kW$].$ You expect a minimum rate of return of $5\%($marr$=0\mathrm{.}05).$ Explain how you would convince the owner to purchase the optimum.$($Conversion factors: $1$ gallon $=3785$ cubic centimeters, $1$ cubic meter $=1000$ liters, $1$ hour $=3600$ seconds$)$

$(50$ points$)$ Determine what the price of fuel in $[$$$/$m$3]$ needs to be in order to receive a $25\%$ return on investment when purchasing a car with an optimum efficiency of $40\%.$ The budget car has an efficiency of $20\%.$ The cars operate on natural gas, where R$=0\mathrm{.}5182[$kJ$/$kg$-$K$],$ and with a lower heating value of $50,050[$kJ$/$kg$].$ The fuel enters the engine at $100[$kPa$]$ and $300[$K$].$ The engine has a maximum net power output of $100[$kW$].$ Based on initial cost quotes, the cost coefficient is $$100/$kW and the exponent is $1\mathrm{.}7.$ The adiabatic flame temperature $($TH$)$ is $2,000[$K$]$ and the ambient temperature $($TL$)$ is $300[$K$].$ Over its $15$ year life, the car overcomes an average force of $1,500[$N$]$ while traveling $20,000[$kilometers each year$].$ Tip: During your calculations, use algebra to solve for operating costs from the definition of cost per unit of fuel and substitute in the equation for savings.

$(25$ points$)$ Should a utility add a regenerator to their gas turbine?$$ The regenerator will add $$50$ million dollars to the initial cost of the gas turbine.$$ The regenerator will increase the thermal efficiency of the cycle from $35\%$ to $55\%.$ The net power output for the application is $100$ megawatts for $8,000$ hours each year. The fuel is natural gas which is delivered to fuel meter at $100[$kPa$]$ and $300[$K$].$ The cost of fuel is expected to be $0\mathrm{.}15[$$$/$m$^3]$ over the $40$ year life of the turbine.$$ The utilities minimum acceptable rate of return is $15\%.$