$$ f^{\prime}(x)=7.18 x-30.8 $$ What are the units of measurement of $$ f^{\prime}(x) $$ ? manatee deaths per year boats per year manatee deaths per 100,000 boats 100,000 boats per manatee deaths (b) Is $$ f^{\prime}(x) $$ increasing or decreasing with increasing $$ x $$ ? increasing (c) Compute and interpret $$ f^{\prime}(8) $$. $$ f^{\prime}(8)=26.64 $$ At a lével of 800,000 boats, the number of manatee deaths is increasing at a rate of $$ 111,416.32 $$ $$ x $$

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**(a)**  
Let $$ f(x) $$ be the number of manatee deaths per year as a function of the number of boats (with the independent variable scaled so that one unit corresponds to 100,000 boats). Then  
$$
f'(x)=7.18x-30.8
$$  
has units of  
$$
\frac{\text{(manatee deaths per year)}}{\text{(100,000 boats)}}.
$$  
Thus, the correct answer is: **manatee deaths per 100,000 boats**.

**(b)**  
Since  
$$
f'(x)=7.18x-30.8
$$  
is a linear function of $$ x $$ with a positive coefficient $$ 7.18 $$ (the slope of the derivative), it increases as $$ x $$ increases.  
The answer is: **increasing**.

**(c)**  
To compute $$ f'(8) $$:
$$
f'(8)=7.18(8)-30.8=57.44-30.8=26.64.
$$  
Since $$ x=8 $$ corresponds to 800,000 boats (because each unit represents 100,000 boats), the interpretation is that when there are 800,000 boats, an additional 100,000 boats would cause the annual number of manatee deaths to increase by approximately $$ 26.64 $$ deaths.

Thus, $$ f'(8)=26.64 $$ means that at a level of 800,000 boats, the annual manatee deaths are increasing at a rate of about $$ 26.64 $$ deaths per additional 100,000 boats.
**(a)** The units of $$ f^{\prime}(x) $$ are **manatee deaths per 100,000 boats**. **(b)** $$ f^{\prime}(x) $$ is **increasing** with increasing $$ x $$. **(c)** $$ f^{\prime}(8)=26.64 $$ At a level of 800,000 boats, the number of manatee deaths is increasing at a rate of approximately 26.64 deaths per additional 100,000 boats.

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