The irrigation system a farmer uses $($Option A$)$ cost $$10,000$ eight years ago. It will last another $25$ years without additional investment. With that system, he produces crops valued at $$3,000$ per year at a cost of $$1,000$ per year. $$

A new system $($Option B$)$ would cost $$15,000$ to install, but would increase production to $$7,000$ per year. Operating cost would be $$2,500$ per year. The farmer would have to refurbish the new system tweleve years after installation at a cost of $$5,000.$

Assume that investment in the new system $($Option B$)$ occurs at the start of the first year, that revenue and operating cost occur at the end of each year and do not change over the twenty$-$five years, and that both systems have a salvage value of $$1,000$ at the end of $25$ years. $$

Assume a $6\%$ discount rate. $$

QUESTIONS:

$(1)$ How should you set up this problem to account for the fact that the existing system is $8$ years old? Should the original investment be discounted by $8$ years to show this? Attached is an example that assumes that capital investment occurs at the end of Year $1.$ How should that be revised? $$

$(2)$ Calculate the Net Present Value for Option A and Option B$.$

$(3)$ Calculate the Benefit$-$Cost Ratio for Option A and Option B$.$

$(4)$ Should the farmer replace his existing system? Why or why not?