A precision lathe costs $17,000 and will cost $27,000 a year to operate and maintain. If the discount rate is 12% and the lathe will last for 5 years, what is the equivalent annual cost of the tool?

The equivalent annual cost of the precision lathe, considering its initial cost, annual operational cost, its expected lifespan, and a discount rate of 12%, is approximately $107,446.24 per year.

Explanation:

The equivalent annual cost of the tool can be calculated by adding both the upfront cost and the operation and maintenance costs over the lifetime of the lathe, and then distributing it evenly over those 5 years. But since the value of money also depends on time, we have to account for the discount rate.

The formula to calculate the Present Value of an Annuity (which is a series of equal payments made at equal intervals) is PV = PMT / (r * (1 - (1 + r) ^ -n)), where PMT is the payment per period (in this case, the annual operational cost of $27,000), r is the discount rate (12% or 0.12), and n is the number of periods (5 years).

The PV of the operational costs becomes $27,000 / (0.12 * (1 - (1 + 0.12) ^ -5)) = $27,000 / (0.12 * (1 - 0.5674)) = $27,000 / 0.0519 = $520,231.21.

Now, to get the equivalent annual cost, add the initial cost of $17,000 to the PV of the operational costs to get a total cost of $537,231.21.

Finally, distribute this total cost over the 5-year lifespan of the lathe: $537,231.21 / 5 = $107,446.24.

So, the equivalent annual cost of the lathe, considering operational costs and a discount rate of 12%, is approximately $107,446.24.

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