Step 1 :The problem is asking for a function that calculates the earnings of the plant per man-hour after t years. The initial earning is $80 per man-hour and it increases by 5% each year. This is an exponential growth problem, where the amount increases by a certain percentage each year.
Step 2 :The general formula for exponential growth is \(A = P(1 + r/n)^{nt}\), where P is the principal amount (initial amount), r is the annual interest rate (in decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
Step 3 :In this case, the initial amount P is $80, the rate r is 5% or 0.05, and the interest is compounded once per year (n=1).
Step 4 :So the function becomes \(A(t) = 80(1 + 0.05/1)^{1*t} = 80(1 + 0.05)^t\).
Step 5 :\(\boxed{A(t) = 80(1 + 0.05)^t}\) is the function that gives the amount $A(t)$ that the plant earns per man-hour $t$ years after it opens.