Final answer:
The equivalent annual worth of the bridge at an interest rate of 10% per year is $39,260,948.94.
Step-by-step explanation:
To determine the equivalent annual worth of the bridge, we need to calculate the present value of all the costs over the lifetime of the bridge. The bridge must be resurfaced every 5 years for a period of time, after which it is expected to be permanent. Let's assume the bridge will last for 50 years.
First, we calculate the present value of the initial cost of $30 million at an interest rate of 10%. Using the formula for present value of a single amount, we get:
Present Value of Initial Cost = $30,000,000 / (1 + 0.10)^0 = $30,000,000
Next, we calculate the present value of the resurfacing costs of $1 million every 5 years. We do this for each 5-year period and then sum up the present values using the formula for an annuity:
Present Value of Resurfacing Costs = (PMT / (1 + i)^n) + (PMT / (1 + i)^(2n)) + ... + (PMT / (1 + i)^(kn))
where PMT is the resurfacing cost, i is the interest rate, n is the number of compounding periods, and k is the number of times the cost is incurred. In this case, k = 10 (50 years / 5 years) and PMT = $1,000,000. Evaluating the sum, we get:
Present Value of Resurfacing Costs = ($1,000,000 / (1 + 0.10)^1) + ($1,000,000 / (1 + 0.10)^2) + ... + ($1,000,000 / (1 + 0.10)^10) = $8,991,080.26
Finally, we calculate the present value of the annual inspection and operating costs of $50,000. Using the formula for present value of an ordinary annuity, we get:
Present Value of Annual Inspection and Operating Costs = PMT * [(1 - (1 + i)^-n) / i] = $50,000 * [(1 - (1 + 0.10)^-50) / 0.10] = $269,868.68
To determine the equivalent annual worth, we sum up the present values of the initial cost, resurfacing costs, and annual inspection and operating costs:
Equivalent Annual Worth = $30,000,000 + $8,991,080.26 + $269,868.68 = $39,260,948.94