The problem asks us to find the model $f(x)$ that represents the number of employees in the solid waste management industry, given the rate of change function and initial conditions.

Given:

• The rate of change of employees, $g(x) = \frac{53.44}{x+1}$, where $x$ is the number of years since 1980, and $g(x)$ is the rate of change in thousands of employees per year.
• $g(x)$ is defined for $0 \leq x \leq 16$, meaning this model covers the years from 1980 to 1996.
• There were 209.5 thousand employees in 1990, which corresponds to $x = 10$ (since $x = 10$ represents 1990, i.e., 10 years after 1980).

Step 1: Finding the Total Number of Employees Function

We need to recover the model $f(x)$, which gives the total number of employees in the industry over time.

Since $g(x)$ is the rate of change of employees, we can express $f(x)$ as the integral of $g(x)$. Thus:

$f(x) = \int g(x) \, dx = \int \frac{53.44}{x + 1} \, dx$

This is a straightforward integration:

$f(x) = 53.44 \ln(x + 1) + C$

where $C$ is the constant of integration that we need to determine using the initial condition.

Step 2: Using the Initial Condition

We know that in 1990 (i.e., $x = 10$), there were 209.5 thousand employees. So we can substitute this into the equation to find $C$:

$f(10) = 53.44 \ln(10 + 1) + C = 209.5$

Simplifying:

$53.44 \ln(11) + C = 209.5$

Using $\ln(11) \approx 2.3979$:

$53.44 \times 2.3979 + C = 209.5$

$128.09 + C = 209.5$

Solving for $C$:

$C = 209.5 - 128.09 = 81.41$

Step 3: Writing the Final Model

Now that we have $C = 81.41$, the total number of employees $f(x)$ is:

$f(x) = 53.44 \ln(x + 1) + 81.41$

This is the model that represents the number of employees in the solid waste management industry in thousands from 1980 to 1996.

Would you like more details or have any questions about this? Here are some related questions you can explore:

1. How do you interpret the behavior of the function $g(x)$ as $x$ increases?
2. Can you predict the number of employees in 1996 (when $x = 16$) using the model $f(x)$?
3. How would the model change if the initial number of employees in 1990 were different?
4. What does the constant $C$ represent in the context of this problem?
5. What would the model look like if the rate of change function were different, such as $g(x) = \frac{a}{x + b}$?

Tip: Always check the units of your variables to ensure consistency throughout the calculations. In this case, $f(x)$ and $g(x)$ are in thousands of employees, so your answers should reflect that.