$8.$ A business has two investment choices. Expected cash flows for each alternative are as follows:

Project# $1$ requires an immediate cash outlay of $$78,500$ and is expected to provide a single cash inflow of$$215,600$ at the end of$7$ years.

Project #$2$ requires an immediate cash outlay of $$85,600$ and is expected to provide cash inflows of $$21,950$ at the end of every year for the next $7$ years.

The required rate of return on the investment $($hurdle rate$)$ is $15\%$ compounded annually.

a$.$ Find the Internal Rate of Return $($IRR$)$ for each project.

b$.$ Find the Net Present Value $($NPV$)$ for each project.

c$.$ Find the Profitability Index for each project.

d$.$ Given your answers in parts a$)$ and b$),$ which project is preferred? Briefly explain why.