A few other students and I are attempting the first question and are going in circles. We are having a really hard time setting up the problem and are unsure of how to solve it in terms of NPV, IRR, and payback period. If you have any hints or insight that might be helpful, we would really appreciate it.
Do you think you can first solve it without respect to NPV or IRR?
Payback period in capital budgeting refers to the period of time required to recoup the funds expended in an investment, or to reach the break-even point.  For example, a $1000 investment which returned $500 per year would have a two-year payback period. The time value of money is not taken into account. Payback period intuitively measures how long something takes to “pay for itself.” All else being equal, shorter payback periods are preferable to longer payback periods. Payback period is popular due to its ease of use despite the recognized limitations described below.
The term is also widely used in other types of investment areas, often with respect to energy efficiency technologies, maintenance, upgrades, or other changes. For example, a compact fluorescent light bulb may be described as having a payback period of a certain number of years or operating hours, assuming certain costs. Here, the return to the investment consists of reduced operating costs. Although primarily a financial term, the concept of a payback period is occasionally extended to other uses, such as energy payback period (the period of time over which the energy savings of a project equal the amount of energy expended since project inception); these other terms may not be standardized or widely used.