Final answer:
To determine the best option using present worth analysis at a 9% interest rate, calculate the present value of the benefits for each alternative. Use the perpetuity formula for Alternative A and equivalent annual benefits for Alternatives B and C considering their respective lifespans and replacement costs. The alternative with the highest present worth is the best option.
Step-by-step explanation:
To find the best alternative using present worth analysis, we need to calculate the present value (PV) of each alternative at the 9% per year interest rate over the given lifespan of the alternatives. Since Alternative A has an infinite lifespan, its annual benefit of $1,375 will be perpetually received, which means we can use the perpetuity formula to calculate its present worth. The formula for perpetuity is PV = (Annual Benefit / Interest Rate).
For Alternative A:
- Present Worth (PW) = $1,375 / 0.09 = $15,277.78
Alternatives B and C are finite and will be replaced with identical replacements at the end of their useful lives, thus we can consider them as perpetuities as well, but we need to calculate the equivalent annual benefit that takes the replacement cost into account.
For Alternative B (18-year lifespan):
- Capital Recovery Factor (CRF) = (Interest Rate * (1 + Interest Rate)^n) / ((1 + Interest Rate)^n - 1)
- Equivalent Annual Benefit (EAB) = (Annual Benefit - (Initial Cost * CRF))
- Present Worth (PW) = EAB / Interest Rate
For Alternative C (9-year lifespan):
- Repeat the same CRF and EAB calculation with 9 years
- Calculate the PW using the obtained EAB
Once the PW for B and C are calculated, compare the PW of all three alternatives. The alternative with the highest present worth is considered the best financial option.