Question

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.
\[A(t)=\]

Answer 1

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:

\[
A(t) = 80 \cdot (1.05)^t
\]