$4)$ NASA scientists have located a large asteroid that they are certain will collide with Earth in $290$ years, doing catastrophic damage and imposing costs of $$5$ trillion $($$$5,000,000,000,000).$ They were prepared for this contingency, and have already proposed a solution: they want to build and launch a rocket that would reposition the asteroid and prevent the collision with Earth entirely. The rocket would be constructed and launched right away $($the Earth$$s current orbit is such that we are as close to this asteroid as we will ever be prior to the projected collision, so this is our best opportunity$),$ so all of the project$$s costs would be incurred immediately. The scientists have learned valuable lessons from the movie Armageddon and are certain that the project would succeed if undertaken.

a$.$ Using a constant annual discount rate of $3\mathrm{.}5\%,$ and assuming that all of NASA$$s projections are correct, how much should we be willing to pay for this intention $($i$.$e$.,$ what is the present value of the project$$s benefits, which are the same as the costs that would be avoided by the project$)?$

b$.$ Using the time$-$declining discount rate schedule proposed by Moore et al$.,$ and assuming that all of NASA$$s projections are correct, how much should we be willing to pay for this intervention $($i$.$e$.,$ what is the present value of the project$$s benefits, which are the same as the costs that would be avoided by the project$)?$