Problem 20 - College Algebra

Question

Use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}.\) To the nearest whole number, what will the fish population be after 2 years?

Step-by-Step Solution

Answer

The fish population after 2 years will be approximately 269.

Step 1: Understanding the Function

The function given is \( P(t) = \frac{1000}{1 + 9 e^{-0.6 t}} \). This is a modified logistic growth model that describes the population of fish over time \( t \). The constant values within the formula control the initial population size, growth rate, and capacity limit.

Step 2: Substituting the Value of t

To find the population after 2 years, substitute \( t = 2 \) into the equation. This gives us the equation: \( P(2) = \frac{1000}{1 + 9 e^{-0.6 \times 2}} \).

Step 3: Calculating the Exponent

Calculate the exponent: \(-0.6 \times 2 = -1.2\). Substitute this value into the equation to simplify the exponent part: \( P(2) = \frac{1000}{1 + 9 e^{-1.2}} \).

Step 4: Finding the Exponential Value

Use a calculator to find the value of \( e^{-1.2} \), which is approximately \( 0.3012 \). Substitute this back into the equation: \( P(2) = \frac{1000}{1 + 9 \times 0.3012} \).

Step 5: Simplifying the Denominator

Multiply \( 9 \times 0.3012 \) to get approximately \( 2.711 \). Then simplify the denominator: \( 1 + 2.711 = 3.711 \).

Step 6: Calculating the Population

Now substitute the simplified denominator back into the equation: \( P(2) = \frac{1000}{3.711} \). Use a calculator to divide 1000 by 3.711, which results in approximately 269.43.

Step 7: Rounding to the Nearest Whole Number

Finally, round 269.43 to the nearest whole number, which is 269.

Key Concepts

Fish Population Modeling

Fish population modeling is a way to represent the growth of a fish population over time using mathematical equations. In this context, the logistic growth model is commonly used, as it effectively captures the nature of population dynamics. Fish populations start small, grow rapidly, and then stabilize. This stabilizing occurs as the population reaches what is known as the 'carrying capacity'—the maximum number that the environment can sustain indefinitely.
Modeling fish populations is crucial for understanding how fish numbers change in response to different factors:

• Resource availability
• Predation pressure
• Environmental conditions

These models help researchers monitor and manage fisheries, ensuring that fish stocks remain healthy.

Exponential Functions

Exponential functions describe mathematical processes where growth or decay becomes rapid over time. In a general sense, exponential growth can be found in situations where quantities increase suddenly and become much larger than initially anticipated. In the logistic growth model, the exponential term helps simulate how the fish population begins to increase exponentially before leveling off.
In an exponential function, variables are often expressed in the form:
\[ y = a imes e^{bx} \]
where \( a \) is the initial value, \( e \) is Euler's number (approximately 2.718), \( b \) determines the growth rate, and \( x \) represents the time or the factor affecting the increase. Such functions are utilized extensively in various fields beyond biology, such as finance and physics.

Population Growth Equations

Population growth equations like the logistic growth model provide a structured way of predicting changes in population size over time. These equations combine both exponential growth and a slowing effect as the population reaches environmental limits.
In the equation given:
\[ P(t) = \frac{1000}{1 + 9 e^{-0.6t}} \]
- \( 1000 \) indicates the carrying capacity of the fish population—an upper limit set by the environment.
- The term \( 1 + 9 \, e^{-0.6t} \) in the denominator helps account for logistic slowing as the fish population grows.
- The coefficient \( -0.6 \) controls the speed at which the population approaches its carrying capacity.
Such equations are pivotal in ecology for predicting how species populations will change, helping in the strategic planning of conservation efforts.