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We are given the formula for the productivity rate of an employee:

\[
n(t) = 70 - \frac{150}{t + 4}
\]

Where $n(t)$ represents the number of items assembled per hour after $t$ weeks of experience. Employees are paid 20 cents per item they assemble.

### Part a: Expression for Money Earned per Hour $ A(t) $

To find an expression for the amount of money earned per hour $A(t)$ by an employee with $t$ weeks of experience, we need to multiply the number of items $n(t)$ they produce by the amount they are paid per item (20 cents, or \$0.20).

\[
A(t) = n(t) \times 0.20
\]

Substituting the given expression for $n(t)$:

\[
A(t) = \left(70 - \frac{150}{t + 4}\right) \times 0.20
\]

Simplifying this:

\[
A(t) = 0.20 \times \left(70 - \frac{150}{t + 4}\right)
\]

\[
A(t) = 0.20 \times 70 - 0.20 \times \frac{150}{t + 4}
\]

\[
A(t) = 14 - \frac{30}{t + 4}
\]

Thus, the expression for the amount of money earned per hour is:

\[
A(t) = 14 - \frac{30}{t + 4}
\]

---

### Part b: Money Earned in the Long Run (as $ t \to \infty $)

To determine how much money an employee should expect to earn per hour in the long run, we take the limit of $A(t)$ as $t \to \infty$:

\[
\lim_{t \to \infty} A(t) = \lim_{t \to \infty} \left( 14 - \frac{30}{t + 4} \right)
\]

As $t \to \infty$, the term $\frac{30}{t + 4}$ approaches 0. Therefore, the limit is:

\[
\lim_{t \to \infty} A(t) = 14 - 0 = 14
\]

Thus, in the long run, the employee can expect to earn **\$14 per hour**.

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### Related Questions:
1. What does the expression $n(t)$ represent in terms of employee productivity?
2. How does the term $\frac{150}{t + 4}$ influence the number of items produced over time?
3. What is the significance of the 0.20 in the formula for $A(t)$?
4. How would changing the payment per item from 0.20 to 0.25 affect the expression for $A(t)$?
5. How long does it take for the employee’s earnings to get close to \$14/hour?

### Tip:
When analyzing long-term behavior in functions, always consider the limit as $t$ approaches infinity to understand the system's equilibrium or steady state.

When starting a new job at a production facility, employees can be expected to assemble n items per hour after t weeks of work experience, where n(t) = 70 - 150/(t + 4). Employees are paid 20 cents for each item they assemble. a) Find an expression for the amount of money A(t) earned per hour by an employee with t weeks of experience. b) How much money per hour should an employee expect to earn in the long run (as t -> ∞)?

Explore the productivity and earnings of an employee in a production facility with a formula for items produced per hour. Learn how to calculate the earnings over time using algebra and limits, and determine how much an employee can earn in the long run.

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