Problem 91 - Algebra and Trigonometry

Question

The fish population in a certain lake rises and falls according to the formula $$F=1000\left(30+17 t-t^{2}\right)$$ Here \(F\) is the number of fish at time \(t\) , where \(t\) is measured in years since January \(1,2002,\) when the fish population was first estimated. (a) On what date will the fish population again be the same as on January \(1,2002 ?\) (b) By what date will all the fish in the lake have died?

Step-by-Step Solution

Answer

(a) January 1, 2019. (b) December 1, 2020.

Step 1: Initial Population Calculation

To determine the initial fish population on January 1, 2002 (when \( t = 0 \)), plug this value into the formula:\[ F(0) = 1000 \times (30 + 17 \times 0 - 0^2) = 1000 \times 30 = 30000. \] Thus, the initial population is 30,000 fish.

Step 2: Find Population Equal to Initial

We need \( F(t) = F(0) = 30000 \). Set up the equation:\[ 1000(30 + 17t - t^2) = 30000. \] Simplify it to:\[ 30 + 17t - t^2 = 30. \] This simplifies to:\[ 17t - t^2 = 0. \] Factor the equation:\[ t(t - 17) = 0. \] The solutions are \( t = 0 \) or \( t = 17 \).

Step 3: Determine the Date for Equal Population

Since \( t = 0 \) corresponds to January 1, 2002, \( t = 17 \) corresponds to 17 years after January 1, 2002, which is January 1, 2019. Thus, the fish population will again be the same on January 1, 2019.

Step 4: Find When Fish Population Reaches Zero

Set \( F(t) = 0 \) and solve:\[ 1000(30 + 17t - t^2) = 0. \] This simplifies to:\[ 30 + 17t - t^2 = 0. \] Rearrange as:\[ t^2 - 17t - 30 = 0. \] Solve using the quadratic formula where \( a = 1 \), \( b = -17 \), \( c = -30 \):\[ t = \frac{-(-17) \pm \sqrt{(-17)^2 - 4 \times 1 \times (-30)}}{2 \times 1} = \frac{17 \pm \sqrt{289 + 120}}{2}. \] \[ t = \frac{17 \pm \sqrt{409}}{2}. \] Calculate \( \sqrt{409} \approx 20.22 \):\[ t = \frac{17 \pm 20.22}{2}. \] The valid (positive) \( t \) is:\[ t \approx \frac{17 + 20.22}{2} = 18.61. \]

Step 5: Convert Decimal Years to Date

The calculation \( t = 18.61 \) years corresponds to approximately December 1, 2020, when rounding the 0.61 year to about 7 months. Therefore, by December 1, 2020, the fish population will be zero.

Key Concepts

Fish Population Modeling

Fish population modeling is a valuable tool that scientists use to understand and predict changes in fish populations over time. In the provided exercise, the fish population in a lake is modeled using a quadratic equation: \[ F = 1000(30 + 17t - t^2) \] This mathematical model helps us to predict how the number of fish evolves with the passage of time, where \( t \) represents time in years from a starting point.
The model contains components that indicate growth and decline of the population. Here, the positive term \( 17t \) suggests a linear increase in population over time, while the \( -t^2 \) term explains a quadratic decline, indicating the population will not grow indefinitely. Understanding these components allows us to predict critical milestones in population changes, such as when the population returns to its initial size or dwindles to zero.
Population models like this one are essential for conservation efforts, helping decision-makers to plan interventions and manage fish populations sustainably.

Time-Dependent Functions

Time-dependent functions change as time progresses. They are instrumental in modeling real-world scenarios like fish populations, where changes occur over time. In our exercise, the function \[ F(t) = 1000(30 + 17t - t^2) \] expresses the fish population's progression as time passes.
The variable \( t \) (time in years) makes the function time-dependent and allows us to plug in specific time values (e.g., \( t=0 \) for January 1, 2002), to find out the number of fish at any given point in time.
Such functions give insights into specific trends: the presence of distinct time points where population equations intersect interesting marks, like returning to initial size or reaching zero. When dealing with such functions, understanding the role of every term can reveal which factors primarily affect the population over time, making it possible to estimate future population values accurately and plan accordingly.

Population Dynamics

Population dynamics examines how populations change over time. It's driven by biological factors and environmental influences. In the context of our lake scenario, population dynamics are encapsulated within the quadratic equation of the fish population. Growth aspects can be seen from the equation's positive term (e.g., \( 17t \)), indicating potential population increase under favorable conditions. The negative term (e.g., \( -t^2 \)) highlights limitations due to factors like limited resources, predation, or environmental capacity, leading eventually to population decline.

• Biological factors include birth rates and natural death rates.
• Environmental factors may encompass nutrient availability and habitat conditions.

Understanding these dynamics is critical in ecological research and fishery management, allowing us to predict future states of the population. An insight gained can inform conservation strategies ensuring sustainable fisheries and ecosystem stability.