Final answer:
To calculate the present worth of the CNC machining center, find the present value (PV) of the initial cost and the annual operating costs. The present worth is $151,304. The future worth at the end of year 10 is $379,143.
Step-by-step explanation:
To calculate the present worth of the CNC machining center, we need to find the present value (PV) of the initial cost and the annual operating costs for the 10 years. Here's how to do it:
- 1. Calculate the PV of the initial cost using the formula PV = FV / (1 + r)^n, where FV is the future value, r is the interest rate, and n is the number of years. In this case, FV = $75,000, r = 20%, and n = 0. PV = $75,000 / (1 + 0.2)^0 = $75,000.
- 2. Calculate the PV of the annual operating costs for the first 4 years. The operating costs are $20,000 for the first 4 years, increasing by 15% per year. We can use the formula PV = C / (1 + r)^n to calculate the PV for each year. For year 1, n = 1, PV = $20,000 / (1 + 0.2)^1 = $16,667. For year 2, n = 2, PV = $20,000 / (1 + 0.2)^2 = $13,889. For year 3, n = 3, PV = $20,000 / (1 + 0.2)^3 = $11,574. For year 4, n = 4, PV = $20,000 / (1 + 0.2)^4 = $9,645.
- 3. Calculate the PV of the annual operating costs for the remaining 6 years. Since the operating costs increase by 15% per year, we can use a geometric series to calculate the PV. The formula for the sum of a geometric series is PV = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = $20,000 and r = 15%. PV = $20,000 / (1 - 0.15) = $23,529.
- 4. Add up all the PVs to get the present worth. Present Worth = PV of initial cost + PV of annual operating costs for the first 4 years + PV of annual operating costs for the remaining 6 years = $75,000 + $16,667 + $13,889 + $11,574 + $9,645 + $23,529 = $151,304.
Therefore, the present worth of the CNC machining center is $151,304.
To calculate the future worth of the CNC machining center at the end of year 10, we can use the formula FV = PV * (1 + r)^n, where PV is the present value, r is the interest rate, and n is the number of years. In this case, PV = $151,304, r = 20%, and n = 10. FV = $151,304 * (1 + 0.2)^10 = $379,143.
The cash flow diagram for this calculation would show the initial cost as a negative value at year 0, the annual operating costs as negative values for the corresponding years, and the future worth as a positive value at year 10.