a$)$ Determine the present values of the following items for an annual interest rate of $10\%$ and $20\%.$

$($i$)$ $$8,000$ earned $6$ years from now.

$($ii$)$ A payment of $$15,000$ at the end of each year for a period of $10$ years.

$($b$)$ A manufacturing process has the following financial information:

Fixed capital $$15,000,000$

Working capital $$4,500,000$

Salvage value $$2,000,000$

Operating cost $$13,000,000/$yr

Revenue $$20,000,000/$yr

SARE expenses $$2,000,000/$yr

Assume a depreciation lifetime of $7$ years, straight$-$line depreciation, and a plant lifetime of $10$

years with a minimum annual rate of return $($MARR$)$ of $15\%$ and a tax rate of $52\%.$ Show the

corresponding discounted cash flow diagrams when answering the following two questions.

$($i$)$ What is the net present value $($NPV$)$ of the process before taxes?

$($ii$)$ What is the NPV of the process after taxes?

Note: Like the operating cost, SARE expenses are also subtracted from the revenue to

compute the profit that is taxed