ERIE COMMUNITY COLLEGE
TITLE III
Quadratic Equations Project
Interdisciplinary Course Materials
Biology
Course: MT 007 Elementary Algebra I/ MT013 Elementary Algebra I and II
Course Outline Topic:
 Evaluate formulas.
 Perform fundamental operations with polynomials.
 Solve quadratic equations by factoring.
 Perform fundamental operations with algebraic fractions and solve rational equations.
Project Title: Logistic Model of Population Growth
Project description: In this project, students will apply the logistic model of population growth.
Author: Dianna Cichocki
Curriculum Expert: Rosanne Redlinski
Semester Created: Fall 2008
A.
Essential Question When are Quadratic Functions applied in the “real world”?
B.
Introduction
The logistic model of population growth assumes that the growth rate of the population decreases
linearly with population size. In particular, the logistic equation gives the instantaneous rate of change of
a population.
C.
Basic Directions
Students will answer three questions relating to the Logistic Model of Population Growth and Quadratic
Functions.
D.
Things to Learn Before Starting the Project
Students should how to write a quadratic equation in standard form. They should also know how to
solve quadratic equations.
E.
The Project Assignment
In this project we use the Logistic Model of Population Growth to answer three questions.
F.
Student Resources
Formula for Logistic Model of Population Growth (given).
G.
Grading Rubric
suggested - 30 points total
Erie Community College
Title III Grant
Name _____________________________
Interdisciplinary Project (Math/Biology)
LOGISTIC MODEL OF POPULATION GROWTH
The logistic model of population growth assumes that the growth rate of the population decreases linearly with
population size. In particular, the logistic equation gives the instantaneous rate of change of a population (ΔN)
as,
where r > 0 is the growth rate of the population with no competition and α > 0 is the per individual effect of
competition. ΔN represents the rate at which the population is growing/decaying at any instant in time.
Problem 1- Write the logistic equation in standard form.
Which of the following equations represents the logistic equation in standard quadratic form ΔN = aN2 + bN +c?
A. ΔN = - αN2 + r
B. ΔN = rN - α
C. ΔN = -αN2 + 1
D. ΔN = - rαN2 + rN
Problem 2- Find the value of N such that the instantaneous rate of population change is zero.
If the growth rate of a population is 0.5 and the effect of competition is 0.01, what value of N > 0 gives an
instantaneous rate of change of the population equal to zero?
A. N = 100
B. N = 50
C. N = 1
D. N = 2
Problem 3- Use the logistic equation to find the carrying capacity of the environment (K)
The logistic equation,
, is often written as,
where K is carrying capacity of the environment. What must K equal for these two forms of the logistic equation
to be equivalent?
A. K = αN
B. K = 1/α
C.
K = rα
D. K = α/(rN)
Erie Community College
Title III Grant
Answer key:
Problem 1- Write the logistic equation in standard form.
We can expand the logistic equation as follows,
ΔN = r(1 - αN)N = rN - rαN2 = -rαN2 + rN.
Problem 2- Find the value of N such that the instantaneous rate of population change is zero.
In this problem we would like to find the value of N > 0 such that,
ΔN = −rαN2 + rN = −(0.5)(0.01)N2 + 0.5N = −0.005N2 + 0.5N = 0.
We can solve the above equation by factoring as,
N (−0.005N + 0.5) = 0.
Since we are looking for a value of N > 0, we know that the above equation implies,
−0.005N + 0.5 = 0,
or N = 0.5/0.005 = 100.
Problem 3- Use the logistic equation to find the carrying capacity of the environment (K)
We can see what K must equal by setting the new equation equal to the original equation as,
Simplifying the above equation yields,
Thus, the carrying capacity is defined as K = 1/α.
Erie Community College
Title III Grant