Honors Discrete Chapter 10 Test Review Guide

Students must be prepared to use formulas and equations to solve word problems, for values of the population sequence or sums of sequences. Problems on the test will focus on 10.2 – 10.4.

Section 10.2 – Linear Growth Model

EXPLICIT Formula for nth Generation: PN = P0+ d*N
Initial Population, P0 Common Difference, d
SUM of Arithmetic:
A0 = FIRST term of addition, AN-1 = LAST term of Addition, N = Number of Terms added
Section 10.3 – Exponential Growth Model
EXPLICIT Formula for nth Generation: PN = P0(r)N
Initial Population, P0 Common Ratio, r
SUM FORMULA of Geometric:
a = first term of addition, r = common ratio, N = # of terms being added
ANNUAL YIELD:The percent of increase from the beginning to the end of one year.
Find the decimal of the common ratio (r) and turn it into a percent increase statement.
Values Given:; Values Not Given:
COMPOUND INTEREST:
/ P0 = principal or initial investment
N = number of years
i = annual interest rate as decimal
k = number of times compounded in a year
Section 10.4 - Logistic Growth Model
Calculator Use: pN = u(n), pN-1 = u(n – 1), and p0 = u(nMin)
Logistic Equation:
  • r = growth parameter (common ratio)
  • pN = p-value; (Actual Population over Carry Capacity)

Honors Discrete Chapter 10 Test Review Guide

Students must be prepared to use formulas and equations to solve word problems, for values of the population sequence or sums of sequences. Problems on the test will focus on 10.2 – 10.4.

Section 10.2 – Linear Growth Model

EXPLICIT Formula for nth Generation: PN = P0+ d*N
Initial Population, P0 Common Difference, d
SUM of Arithmetic:
A0 = FIRST term of addition, AN-1 = LAST term of Addition, N = Number of Terms added
Section 10.3 – Exponential Growth Model
EXPLICIT Formula for nth Generation: PN = P0(r)N
Initial Population, P0 Common Ratio, r
SUM FORMULA of Geometric:
a = first term of addition, r = common ratio, N = # of terms being added
ANNUAL YIELD:The percent of increase from the beginning to the end of one year.
Find the decimal of the common ratio (r) and turn it into a percent increase statement.
Values Given:; Values Not Given:
COMPOUND INTEREST:
/ P0 = principal or initial investment
N = number of years
i = annual interest rate as decimal
k = number of times compounded in a year
Section 10.4 - Logistic Growth Model
Calculator Use: pN = u(n), pN-1 = u(n – 1), and p0 = u(nMin)
Logistic Equation:
  • r = growth parameter (common ratio)
  • pN = p-value; (Actual Population over Carry Capacity)

Honors Discrete Chapter 10 Test Practice Problems

SECTION 10.2 – LINEAR GROWTH MODELS

For each of the following linear growth model,

  • Find an explicit description for the population
  • Find the 9th generation of the population, P9

1)P0 = 718 and d = 32

2)3, 19, 35, 51, …

3)P5 = 97 and P6 = 86

4)P13 = 217 and P14 = 228

5)P6 = 165 and P15 = 318

6)P21 = 1896 and P90 =4449

For each of the following linear growth model, find the given sum

7)8 + 14 + 20 + 26 + 32 + 38 + 44 + 50

8)P0 = 115 and d = 83;

  1. P0 + P1 + P2 + … + P29
  1. P21 + P22 + … + P30 + P31

9)For 10 terms, 11 + 27 + 43 + 59 + …

10)For 50 terms, 37 + 44 + 51+ …

11)17 + 29 + 41 + … + 1157

12)8895 + 8778 + … + 7140

13)P3 = 127 and P16 = 400;

  1. P0 + P1 + P2 + … + P49
  1. P2 + P4 + P6 + … + P22 + P24 + P26

LINEAR GROWTH WORD PROBLEMS:

14)Starting in the year 2000, the number of crimes committed each year in is predicted to grow according to a linear growth model. During the year 2005, Monroe recorded 115 crimes. During the year 2006, Monroe recorded 121 crimes.

  1. How many crimes were committed in 2000?
  1. How many crimes are predicted to occur in 2007?
  1. How many total crimes were committed between 2000 and 2010?
  1. Write an equation that describes the number of crimes in years since 2000.

15)The production of toy trains is scheduled to be increased by 27 trains a month. It costs the company $3.50 to produce each train per month. When production started in cost the company $2800 for all trains.

  1. How many trains were produced initially?
  1. How many trains will be produced in the 5th month of production?
  1. Write an equation that describes the number of trains produced each month.
  1. After one year, how many total trains were produced?

16)A small business sells $15,500 worth of products during its first year of business. The owner has set an annual goal for increased sales at $1250 for 30 years. Assuming the goal is met, find the total sales during the first 12 years this business is in operation.
SECTION 10.3 – EXPONENTIAL GROWTH MODELS

For each of the following exponential growth model,

  • Find an explicit description for terms of the population
  • Find the 9th generation of the population, P9

1)P0 = 12 and r = 3.5

2)3, 6, 12, …

3)3.5, 9.45, 25.515, 68.8905, …

4)P4 = 202.5 and P5 = 303.75

For each of the following exponential growth model, Find given sum

5)r = 2.2 and P0 = 3
P0 + P1 + P2 + … + P10

6)900 + 900(0.75) + 900(0.75)2 + … + 900(0.5)6

7)125 + 150 + 180 + … + 373.248

8)r = 0.5 and P0 = 98304

  1. P0 + P1 + P2 + … + P13
  1. P7 + P8 + … + P15
  1. P3 + P4 + … + P11

EXPONENTIAL GROWTH WORD PROBLEMS:

9)The number of applicants is expected to grow by 35% each year for the next 15 years. If the original number of applicants was 220, then how many applicants are there predicted to be in 10 years?

10)The number of reported cases of a virus is supposed to decay by 15% each year for 10 years. There currently are 1,200,000 cases of the virus. How cases are expected 6 years from now?

11)The number of certain type of bacteria increases at a rate of 20% every year. Suppose there were 3600 bacteria in 2009.

  1. How many bacteria were there in 2007?
  1. How many bacteria will there be in 2012?
  1. Write an equation that describes the number of bacteria per year since 2007.

12)How much interest would you earn on an account with 20% annual interest rate compounded daily that you initially invested $2500 after 3 years?

13) Michaela has an option between two savings accounts. Account #1 she plans to invest $5500 at 20% annual interest rate compounded quarterly and Account #2 she plans to invest $6000 at 15% annual interest rate compounded monthly. Which account should she choose if she only needed to save money for 5 years?

14)What is the annual yield for an account with 10% annual interest compounded quarterly?

SECTION 10.4: LOGISTIC GROWTH MODEL

a. Write TRANSITION RULE and SEED for the logistic growth model

b. Determine the PREDICTED behavior

(Stable Equilibrium, Cycle Pattern, Attracting Point or Chaos)

c. If possible, provide the VALUE(S) of the predicted behavior

1) The initial p-value is p0 = 0.3 and r = 3.2.

2) The population of students is currently 1200. The maximum capacity of the school is actually 2000. The growth parameter of students is 1.5