Based on the analysis, should the company open the mine?
My answer is Yes
Using the payback period method, it can help to assess the project worthiness. It is the amount of time the company should recover its initial investment. However, it must be noted that one of the drawbacks of using this method is that it ignores the time value of money and any cash outflows that will happen after the payback period. Since the internal rate of return is greater than the required return of capital for the mining investment, it will mean the company will be earning more than what it needs. This can also be confirmed by the positive net present value arrived at during the computations. I suggest that the company should go forward with the investment.
Year Cash flow
0 -$725,000,000
1 $90,000,000
2 $135,000,000
3 $180,000,000
4 $245,000,000
5 $232,000,000
6 $170,000,000
7 $120,000,000
8 $95,000,000
9 -$80,000,000

A. Payback Period Calculation
Payback period = number of year before initial investment is paid off:

Year 0 = -750 year 1 = -750 + 140 = -610 year 2 = -750 + 140 + 180 = -430 year 3 = -750 + 140 + 180 + 210 = -220 = -750 + 140 + 180 + 210 + 230 = +10

220/230 = 0.956
So the total payback period is: 3+0.956 or 3.956 years
Based on your analysis should the company open the mine?
Since the NPV of the mine is positive, the company should open the mine.
B. Internal Rate of Return 23.73% 23.93%
C. Modified Internal Rate of Return 16.21% 16.37%
D. Net Present Value $ 171,141,294.31 $176,550,444.68

Bonus question: Most spreadsheets do not have a built-in formula to calculate the payback period. Write a VBA script that calculates the payback period for a project.
There are many possible variations on the VBA code to calculate the payback period. Below