Answer:
To find the value of \( t \) when the population of Arctic flounder in the Atlantic Ocean is a certain value \( P(t) \), you need to solve the equation:
\[ P(t) = \frac{11t + 24}{0.4t^2 + 3} \]
Let's say you want to find \( t \) when the population is a specific value, \( P_0 \). The equation becomes:
\[ P_0 = \frac{11t + 24}{0.4t^2 + 3} \]
Now, you need to isolate \( t \) on one side of the equation. Here's how you can do it step by step:
1. Multiply both sides of the equation by \( 0.4t^2 + 3 \) to get rid of the denominator:
\[ P_0(0.4t^2 + 3) = 11t + 24 \]
2. Distribute \( P_0 \) on the left side:
\[ 0.4P_0t^2 + 3P_0 = 11t + 24 \]
3. Move all the terms involving \( t \) to one side of the equation and all the constants to the other side:
\[ 0.4P_0t^2 - 11t + 3P_0 - 24 = 0 \]
4. This is now a quadratic equation in terms of \( t \). You can solve it using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a = 0.4P_0 \), \( b = -11 \), and \( c = 3P_0 - 24 \). Plug these values into the quadratic formula and solve for \( t \).
\[ t = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(0.4P_0)(3P_0 - 24)}}{2(0.4P_0)} \]
Simplify further if needed. This will give you the values of \( t \) for which the population of Arctic flounder in the Atlantic Ocean is equal to \( P_0 \).
Explanation: