Questions: The population in millions of arctic flounder (a type of fish) in the Atlantic Ocean is modeled by the function P(t)=(7t+96)/(0.8t^2+1), where t is measured in years. 1. What is the initial flounder population? Number million 2. What is P'(8), rounded to 2 decimal places? Units should be consistent with those defined in the initial problem. P'(8)= Number I

The population in millions of arctic flounder (a type of fish) in the Atlantic Ocean is modeled by the function P(t)=(7t+96)/(0.8t^2+1), where t is measured in years.
1. What is the initial flounder population? Number million
2. What is P'(8), rounded to 2 decimal places? Units should be consistent with those defined in the initial problem.
P'(8)= Number I

Transcript text: The population in millions of arctic flounder (a type of fish) in the Atlantic Ocean is modeled by the function $P(t)=\frac{7 t+96}{0.8 t^{2}+1}$, where $t$ is measured in years. 1. What is the initial flounder population? Number million 2. What is $P^{\prime}(8)$, rounded to 2 decimal places? Units should be consistent with those defined in the initial problem. \[ P^{\prime}(8)=\text { Number } I \]

Solution

Solution Steps

Step 1: Finding the Initial Population

The initial population is calculated by substituting $t=0$ into the general form $P(t) = \frac{at + b}{ct^2 + d}$, which simplifies to $P(0) = \frac{b}{d} = \frac{96}{1}$. Therefore, the initial population of arctic flounder in the Atlantic Ocean is 96 million.

Step 2: Finding the Rate of Change at a Specific Time

Final Answer:

The initial population of arctic flounder in the Atlantic Ocean is 96 million. $P'(8)$ means that at year 8, the population is changing at a rate of -0.58 million flounder per year, indicating a declining population at that time.

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