Step 1 :Given the function \(f(x) = 68.79 e^{0.39x}\), where \(f(x)\) is the cost in dollars per year at time \(x\), and \(x\) is the number of years measured from the beginning of the year 1996.
Step 2 :We need to find the total increase in costs during the next 8 years, beginning in 1996.
Step 3 :To do this, we calculate the cost at the beginning and at the end of 8 years.
Step 4 :The cost at the beginning of 1996 is \(f(0) = 68.79\).
Step 5 :The cost at the end of 2003 is \(f(8) = 1557.8444556540358\).
Step 6 :The total increase in costs is the cost at the end minus the cost at the beginning, which is \(1557.8444556540358 - 68.79 = 1489.0544556540358\).
Step 7 :Rounding to the nearest cent, the total increase in costs over the next 8 years is approximately $1489.05.
Step 8 :Final Answer: The total increase in costs is \(\boxed{1489.05}\).