Problem 45 - Precalculus: Functions and Graphs

Question

Deer population A herd of 100 deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. Suppose that the number \(N(t)\) of deer after \(t\) years is given by \(N(t)=-t^{4}+21 t^{2}+100,\) where \(t>0\) (a) Determine the values of \(t\) for which \(N(t)>0,\) and sketch the graph of \(N\) (b) Does the population become extinct? If so, when?

Step-by-Step Solution

Answer

(a) N(t) > 0 for 0 < t < 5. (b) Yes, population becomes extinct at t = 5.

Step 1: Find the roots of N(t) = 0

To find when the deer population becomes zero, set the equation equal to zero: \[-t^4 + 21t^2 + 100 = 0\]. This can be rewritten as a quadratic in terms of \(x = t^2\): \[-x^2 + 21x + 100 = 0\]. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -1\), \(b = 21\), and \(c = 100\) to find the roots.

Step 2: Solve quadratic equation for x

Using the quadratic formula, we have: \[x = \frac{-21 \pm \sqrt{21^2 - 4(-1)(100)}}{2(-1)}\]. Simplify the expression to find \(x\):\[x = \frac{-21 \pm \sqrt{441 + 400}}{-2}\].\[x = \frac{-21 \pm \sqrt{841}}{-2}\].\[x = \frac{-21 \pm 29}{-2}\].Calculate the positive root: \[x_1 = \frac{8}{-2} = -4\] which is not a valid solution for \(t^2\), and the other:\[x_2 = \frac{-50}{-2} = 25\]. Thus, \(t^2 = 25\) gives \(t = 5\).

Step 3: Determine intervals where N(t) > 0

N(t) changes sign at the roots. The expression is positive between the roots (since this is a downward opening parabola in terms of \(t^2\)). Based on calculations from before, try values like \(t = 1\) or \(t=3\) and confirm the switch in sign across \([0,5)\). Since N(t) is a polynomial, it will be greater than zero in this interval excluding the point where it crosses the t-axis. Thus, \(N(t) > 0\) for \(0 < t < 5\).

Step 4: Sketch the Graph N(t)

Plot the curve \(N(t)=-t^4 + 21t^2 + 100\) using the calculated information. The graph starts above the x-axis and curves downward, showing growth and then decline of the population reaching zero at \(t = 5\). The graph is symmetrical about the y-axis due to its even powers in the equation.

Step 5: Analyze extinction condition

To determine if the population becomes extinct, check whether there exists a time \(t\) where \(N(t) = 0\) and does not recover beyond this point. From Step 2, \(t = 5\) results in end of population given \(t > 0\). Beyond \(t=5\), the polynomial results in negative values which are not feasible population sizes. Hence, the population does become extinct at \(t = 5\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of variables multiplied by coefficients. In the case of deer population modeling, we are dealing with a polynomial function that describes how the population changes over time based on a formula.
The polynomial given in this exercise is:

In our polynomial, the highest exponent is 4, which is why it's called a quartic function. Such polynomials are crucial for understanding complex population dynamics as they capture potential rises and falls over time. The coefficients and degree directly influence the shape and behavior of the function.

Finding Roots

Finding the roots of a polynomial function means determining the values of the variable for which the function equals zero. These values, also known as "zeros" or "solutions," are essential for understanding how a population modeled by the polynomial might behave over time.

To find the roots of our population model, set:

• • To solve the problem more easily, we make a substitution where
    • > x = t^2 , transforming the original equation into a quadratic one:
      • > -x^2 + 21x + 100 = 0. This substitution helps simplify the process by reducing the degree of the polynomial.
  Finding the roots of a polynomial is fundamental for analyzing when populations drop to zero or become extinct. It is through this method that we can identify critical turning points in population dynamics.

Quadratic Formula

The quadratic formula is a tool for solving quadratic equations of the form ax² + bx + c = 0. In deer population problems, it helps determine when the population might reach zero. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

We applied this formula to our substituted equation,

• -x² + 21x + 100 = 0, using values
  • > a = -1
  • > b = 21
  • > c = 100
  to find the roots:
  • > x = \( \frac{-21 \pm \sqrt{841}}{-2} \) leading to \(x_1 = -4\) (invalid for \(t^2\)) and \(x_2 = 25\), thus \(t = 5\).
  
  
  The quadratic formula is an invaluable component in any mathematical toolbox, especially when dealing with real-world scenarios like population dynamics. It allows us to solve for exact time points when populations meet certain conditions, such as extinction.

Graphing Functions

Graphing functions is a visual way to understand how mathematical models predict changes over time. For the deer population, graphed using the function:

• • \(N(t) = -t^4 + 21t^2 + 100\), we analyze the growth and decline phases by examining the graph.

The graph offers a clear snapshot by tracking:

• Increasing intervals, showing growth in deer numbers.
• The peak population, which signals the highest point before decline.
• Zero crossings, crucial for identifying when the population becomes extinct, as noted at \(t = 5\).

The graph is symmetric with respect to the y-axis due to equal powers on \(t\), further understanding the balanced rise and fall in population based on available resources. Visual graphs are intuitive tools that complement algebraic solutions, making it easier to grasp how populations fluctuate.