Final answer:
To find the perpetual equivalent annual maintenance cost for a coal mining monitoring system, calculations involve the present value of a growing annuity for the increasing maintenance costs and a perpetuity for the constant costs thereafter, which are then converted into an equivalent annuity at a 10% interest rate.
Step-by-step explanation:
To determine the perpetual equivalent annual maintenance cost for a monitoring system in a coal mining operation, we can use the present value of an annuity and the growth of an annuity formula. First, we calculate the present value of the expected maintenance costs from year 3 to year 6, which are increasing by 5% each year. Then, since the costs are expected to remain constant after year 6 through the system's 10-year life, we calculate the present value of these costs as a perpetuity from year 6 onwards. Finally, we can determine the equivalent annual cost by converting this total present value into an annuity over the infinite lifespan of the system replacements.
Mathematically, the calculation proceeds by first finding the present value at year 2 (one year before the first cost is incurred) of the costs from year 3 to 6 using the formula for the present value of a growing annuity. Then calculating the present value at year 2 of the perpetuity starting from year 6 using the perpetuity formula with costs growing at a rate of 0% (since they level off). To convert these into an equivalent annual cost, the result is treated as a present value of an ordinary annuity with the interest rate i=10% over an infinite time horizon.