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Chapter 2: Problem 8

The growth \(G\) of a population of lower organisms over a day is a function of the population size \(n\) at the beginning of the day. If both \(n\) and \(G\) are measured in thousands of organisms, the formula is $$ G=-0.03 n^{2}+n $$ a. Make a graph of \(G\) versus \(n\). Include values of \(n\) up to 40 thousand organisms. b. Calculate \(G(35)\) and explain in practical terms what your answer means. c. For what two population levels will the population grow by 5 thousand over a day? d. If there is no population to start with, of course there will be no growth. At what other population level will there be no growth?

Short Answer

Expert verified
a. Plot the function up to \( n = 40 \). b. \( G(35) = -1.75 \), the population decreases by 1.75 thousand. c. Populations that grow by 5 thousand: \( 6.2 \) and \( 26.2 \). d. No growth at \( n = 33.33 \).

Step by step solution

01

Graph Setup

To make a graph of \( G = -0.03n^2 + n \) versus \( n \), we need to evaluate \( G \) for values of \( n \) ranging from 0 to 40. Create a table of values where you substitute each integer \( n \) from 0 to 40 into the equation and compute \( G \).
02

Create Graph

Using the calculated values from Step 1, plot the points \((n, G)\) on a graph. The x-axis represents the population size \( n \) (in thousands) and the y-axis represents the growth \( G \). Connect the points to observe the shape of the graph.
03

Calculate G(35)

Substitute \( n = 35 \) into the formula: \[ G(35) = -0.03(35)^2 + 35 \]. Simplify this to find \( G(35) \). Calculate \( 35^2 = 1225 \), then \( -0.03 \times 1225 = -36.75 \), and finally add 35 to get \( -36.75 + 35 = -1.75 \). Thus, \( G(35) = -1.75 \).
04

Interpret G(35) Result

The result \( G(35) = -1.75 \) thousand means that a population of 35 thousand organisms would decrease by 1.75 thousand over the day. This is a decline rather than growth.
05

Solve for Population Levels with G = 5

Set the equation \( G = -0.03n^2 + n \) to 5 and solve for \( n \): \[ 5 = -0.03n^2 + n \]. Rearrange to get \( -0.03n^2 + n - 5 = 0 \). This is a quadratic equation of the form \( ax^2 + bx + c = 0 \) with \( a = -0.03 \), \( b = 1 \), and \( c = -5 \). Use the quadratic formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
06

Solve Quadratic Equation

Substitute into the quadratic formula: \( n = \frac{-1 \pm \sqrt{1^2 - 4(-0.03)(-5)}}{2(-0.03)} = \frac{-1 \pm \sqrt{1 - 0.6}}{-0.06} = \frac{-1 \pm \sqrt{0.4}}{-0.06} \). Calculate \( \sqrt{0.4} \approx 0.632 \). Solve for \( n \): \( n_1 \approx \frac{-1 + 0.632}{-0.06} \) and \( n_2 \approx \frac{-1 - 0.632}{-0.06} \). The solutions are \( n_1 \approx 6.2 \) and \( n_2 \approx 26.2 \).
07

No Growth Calculation

Set \( G = 0 \) in the formula \( G = -0.03n^2 + n \) to find when there is no growth. Solve \( 0 = -0.03n^2 + n \), which simplifies to \( 0 = n(-0.03n + 1) \). This gives solutions \( n = 0 \) or \( n = 33.33 \) (the result of \( n = \frac{1}{0.03} \)).
08

Interpret Growth Findings

At \( n = 33.33 \) thousand, the growth is zero, indicating that a population of 33.33 thousand organisms will stay the same size over the day.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
A quadratic function is a type of polynomial function of degree two, typically expressed in the form \( f(x) = ax^2 + bx + c \). In this formula,
  • \( a \), \( b \), and \( c \) are constants
  • \( a \) is non-zero
  • The graph of a quadratic function is a parabola
In the context of population growth, the function \( G = -0.03n^2 + n \) represents a quadratic function where:
  • \( a = -0.03 \)
  • \( b = 1 \)
  • \( c = 0 \)
This specific quadratic function models how the population of a species grows with respect to its size \( n \). The parabola will open downwards because \( a \) is negative, indicating that there will be a maximum point after which the growth starts to decrease. Quadratic functions are useful in modeling biological processes where some upper limit or threshold effects the behavior.
Graphing Techniques
Graphing is a fundamental tool in visualizing the behavior of a function over a range of inputs. To graph the quadratic function \( G = -0.03n^2 + n \), we analyze how growth, \( G \), changes as the population size, \( n \), grows.
  • Start by calculating \( G \) for different values of \( n \), from 0 up to 40 (thousand)
  • Create a table of values for easier plotting
  • Plot these \( (n, G) \) points on a coordinate plane where the horizontal axis (x-axis) represents \( n \) and the vertical axis (y-axis) represents \( G \)
By connecting these points, you'll obtain a parabola. The shape is informative, showing where the growth peaks and when it starts to decline.
Understanding graphing techniques helps in predicting future population sizes and planning ecological interventions. This visual method highlights both the potential growth and decline in populations, guiding strategic decisions.
Population Dynamics
Population dynamics refers to the way populations change in size and composition over time. This involves understanding growth rates and the factors that influence them.

In the formula \( G = -0.03n^2 + n \), population growth is not a simple linear relationship鈥攔ather, it depicts how growth depends on the current population size.
  • As \( n \) increases, \( G \) first increases, reaches a peak, and then decreases, highlighting the presence of a carrying capacity.
  • The quadratic nature indicates that very large populations may experience overcrowding, resource depletion, or other negative effects that lead to reduced growth.
This model sheds light on real-world dynamics by suggesting that populations have optimal sizes for growth鈥攏either too small nor too large鈥攁t which steady and sustainable development can be achieved.
Algebraic Solutions
Algebra provides the tools to solve equations, especially when calculating specific outcomes like in this exercise.
  • The quadratic equation \( 5 = -0.03n^2 + n \) can be set up to find specific population sizes \( n \) where growth \( G = 5 \) thousand.
  • Rewriting it in the form \( ax^2 + bx + c = 0 \) allows the use of the quadratic formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
This computation reveals two population sizes: \( n_1 \approx 6.2 \) thousand and \( n_2 \approx 26.2 \) thousand, where the target growth occurs.

Solving these kinds of algebraic problems provides insight into biological constraints and optimum growth conditions. It also aids in predicting changes over time, which is crucial for conservation and management efforts.

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Most popular questions from this chapter

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