Exponential expressions word problems (algebraic)
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A manufacturing plant earned $\$ 80$ per man-hour of labor when it opened. Each year, the plant earns an additional $5 \%$ per man-hour.
Write a function that gives the amount $A(t)$ that the plant earns per man-hour $t$ years after it opens.
\[
A(t)=
\]
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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.