Use the following references below to solve a$,$b$,$c and d using the image to guide the problem.

For a reference LWR: assume a thermal efficiency is $33\%,$ power $1150$ MWe, uranium is $$70/$lb$.$ U$3$O$8,$ SWU is $$120/$kgSWU$,$ conversion is $$40/$kgU$,$ fabrication costs are $$400/$kgU$,$ losses in conversion $0\mathrm{.}15\%,$ losses in fabrication $0\mathrm{.}1\%,$ tails assay of $0\mathrm{.}2$wt$\%,$ reference core contains $90$ tonne fuel with enrichment of $4\mathrm{.}2\%.$ An average economical grade of uranium ore is $0\mathrm{.}2\%.$ Each fuel assembly contains $500$kg of fuel. The reactor operates on a $3-$batch equilibrium cycle of $18$ months which includes $17$ months at power including $3-$day startup and $1$ day for shutdown.

Data for recycling and disposal: Price of reprocessing $($$$/$kg used fuel$)=1000,$ Price of MOX fabrication $($$$/$kg$)=500,$ Price of DU $($$$/$kg$)=5,$ used fuel is $0\mathrm{.}93$ kg used fuel$/$kg fresh fuel, used fuel has $6\mathrm{.}0$ g fissile Pu$/$kg used fuel, fissile Pu has a reactivity equivalency of $0\mathrm{.}8$ g U$235$equivalent$/$g fPu$),$ loss in reprocessing$=0\mathrm{.}005$; losses in fabrication same as UO$2$ fuel; Table $1$ shows information on reactor grade plutonium; dry casks can hold $32$ spent fuel assemblies and costs $$1\mathrm{.}2$M to purchase and load. Financial$/$economic data: annual interest rate is $4\%,$ interest is compounded monthly

Evaluate an equivalent MOX fuel for LWR fabricated from natural uranium and reactor grade plutonium from PWR with a burnup of $53,000$ MWD$/$MTU:

a$.$ Calculate the total plutonium and fissile Pu loading of a MOX fuel that is equivalent to the UO$2$ fuel referenced in the given data above $($report as kg Pu per kg fuel and kg fissilePu per kg fuel respectively$).$

b$.$ What would be the cost per kg of this MOX fuel?

c$.$ Considering the time value of money, calculate the cost of the MOX fuel $($$$/$kg$)$ and the cost of a reload for the reactor at the point it is loaded into the reactor $($one$-$third MOX, two$-$thirds UO$2).$ Assume reprocessing costs are paid $30$ months in advance and fabrication costs paid $24$ months in advance.

d$.$ Assuming approximately $100,000$t used fuel in the US with the specifications given above, how long could you fuel the $93$ current reactors in the US$?$ Assume each has the same power as the reference LWR and computed capacity factor. Ignore the restrictions on the amount of MOX fuel that can be utilized in a reactor at any given time. The problem is asking for an evaluation of fuel energy value in comparison to demand.