To model the earnings per man-hour for the manufacturing plant over time, we can use an exponential growth function, since the earnings increase by a constant percentage each year.
The initial earnings when the plant opened is \( A(0) = 80 \) dollars per man-hour, and it increases by \( 5% \) per year.
To express the function, we can use the formula for exponential growth:
\[ A(t) = A(0) \cdot (1 + r)^t \]
where:
- \( A(0) \) is the initial amount earned per man-hour,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the number of years after the plant opens.
Substituting the values we have:
- \( A(0) = 80 \),
- \( r = 0.05 \) (since \( 5% = 0.05 \)),
we get:
\[ A(t) = 80 \cdot (1 + 0.05)^t \] \[ A(t) = 80 \cdot (1.05)^t \]
Thus, the function that gives the amount \( A(t) \) that the plant earns per man-hour \( t \) years after it opens is:
\[ A(t) = 80 \cdot (1.05)^t \]