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MATH1013 Quizzes

Quiz 4: First order linear difference equations
Question 1 Questions
What are the next four terms in the sequence xn+1 0.4xn 150 = 0, where x6 = 50?
a)
x1 = 170, x2 = 218, x3 = 237.2, x4 = 244.88
 b)
x1 = 210, x2 = 234, x3 = 243.6, x4 = 247.44
c)
x7 = 170, x8 = 218, x9 = 237.2, x10 = 244.88
 d)
x7 = 210, x8 = 234, x9 = 243.6, x10 = 247.44

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
xn+1 = 0.4xn + 150. Since we are given x6, the next 4 terms are x7 = 0.4 50 + 150 = 170, x8 = 0.4 170 + 150 = 218, x9 = 0.4 218 + 150 = 237.2, x7 = 0.4 237.2 + 150 = 244.80.
Choice (d) is incorrect
What are the next four terms in the sequence un+2 5un+1 + 6un = 20, where u1 = 4 and u2 = 9?
a)
u2 = 9, u3 = 41, x4 = 171, u5 = 629
 b)
u3 = 41, x4 = 171, u5 = 629,u6 = 2139
c)
u2 = 9, u3 = 14, x4 = 74, u5 = 308
 d)
u3 = 14, x4 = 74, u5 = 308, u6 = 1116

Choice (a) is incorrect
Choice (b) is correct!
un+2 = 5un+1 6un + 20. Since we are given u1 and u2 the next 4 terms are u3 = 5 9 6 4 + 20 = 41, u4 = 5 41 6 9 + 20 = 171, u5 = 5 171 6 41 + 20 = 629, u6 = 5 629 6 171 + 20 = 2139.
Choice (c) is incorrect
Choice (d) is incorrect
Let Cn be the number of people who were on the subscription list of the magazine “UFO today” n years after the beginning of 1995. What is the meaning of C4?
a)
C4 = the number of people who were on the subscription list of the magazine in the year 2000.
 b)
C4 = the number of people who were on the subscription list of the magazine at the beginning of the year 2000.
c)
C4 = the number of people who were on the subscription list of the magazine in 1998.
 d)
C4 = the number of people who were on the subscription list of the magazine at the beginning of 1999.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
C1 is the number of people who were on the subscription list of the magazine at the beginning of 1996.
C2 is the number of people who were on the subscription list of the magazine at the beginning of 1997.
C3 is the number of people who were on the subscription list of the magazine at the beginning of 1998.
C4 is the number of people who were on the subscription list of the magazine at the beginning of 1999.
Suppose Pn = 0.4Pn1 and P1 = 100. What is Pn in terms of n?
a)
Pn = 100(0.4)n1
 b)
Pn = 100(0.4)n
c)
Pn = 100(0.4)n1(0.6)
 d)
Pn = 100(0.6)n1(0.4)

Choice (a) is correct!
If Pn = aPn1 then Pn = an1P1.
Therefore Pn = (0.4)n1100.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Suppose Pn = 0.8Pn1, where n is the number of years since 1960. Determine how long it would take for the population to halve.
a)
It would take until 1964.
 b)
Just over 3 years.
c)
Just over 4 years.
 d)
The answer is negative and makes no sense.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
We wish to solve Pn = 1 2P1 = (0.8)n1P 1, therefore (0.8)n1 = 1 2, ln(0.8)n1 = ln 1 2, (n 1)ln(0.8) = ln2, n 1 = ln2 ln0.8 = 3.106, n = 4.106.
Choice (d) is incorrect
Suppose Pn = 1.04Pn1, where n is the number of years since 1980. Determine how long it would take for the population to double.
a)
It would take until 2008.
 b)
Just over 2 years.
c)
About 17 years and 8 months.
 d)
About 18 years and 8 months.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
We wish to solve Pn = 2P1 = (1.04)n1P1, therefore (1.04)n1 = 2, ln(1.04)n1 = ln2, (n 1)ln(1.04) = ln2, n 1 = ln2 ln1.04 = 17.673, n = 18.673.
Questions 7 and 8 use the same information.
 
The Bureau of Statistics has issued the following annual population information for the town of Agga Agga for the ten years since 1990. The town has an annual birth rate of 4.5% of the population and an annual death rate of 2.5% and there was a nett inflow of 100 people migrating to the town each year.
 
Which of the difference equations below would model the population, Pn, of Agga Agga where n is the number of years since 1990 ?
a)
Pn = 7Pn1 100
 b)
Pn 2Pn1 100 = 0
c)
Pn 1.02Pn1 100 = 0
 d)
Pn = 1.07Pn1 100

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The nett growth rate is 2%. If there was no migration we would have Pn = 1.02Pn1. With the nett migration of 100 people we would have Pn = 1.02Pn1 + 100.
Choice (d) is incorrect
Questions 7 and 8 use the same information.
 
The Bureau of Statistics has issued the following annual population information for the town of Agga Agga for the ten years since 1990. The town has an annual birth rate of 4.5% of the population and an annual death rate of 2.5% and there was a nett inflow of 100 people migrating to the town each year.
 
If the population of Agga Agga was 10 000 in 1990 what was the population to the nearest 10 people in 1995 ?
a)
11240
 b)
11560
c)
11890
 d)
10520

Choice (a) is incorrect
Choice (b) is correct!
If Pn is the population of Agga Agga then P0 = 10000 so we are looking for P5 and
P1 = 10300,P2 = 10606,P3 = 10918.12,P4 = 11236.21,P5 = 11561.21.
Choice (c) is incorrect
Choice (d) is incorrect
Questions 9 and 10 use the same information.
 
A population of Bilbies in western New South Wales is decreasing at a rate of 8% per year. The population has been supplemented by 75 breeding pairs per year from Taronga Zoo starting in 1998.
 
Which of the following difference equations model the population if Pn is the population n years since the population supplement began ?
a)
Pn 0.92Pn1 150 = 0
 b)
Pn 1.08Pn1 75 = 0
c)
Pn 1.08Pn1 150 = 0
 d)
Pn 0.92Pn1 75 = 0

Choice (a) is correct!
The rate of population decrease is 8%, i.e there is only 92% of the population left after each year. If there was no supplement we would have Pn = 0.92Pn1. With the supplement of 150 Bilbies we would have Pn = 0.92Pn1 + 150.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Questions 9 and 10 use the same information.
 
A population of Bilbies in western New South Wales is decreasing at a rate of 8% per year. The population has been supplemented by 75 breeding pairs per year from Taronga Zoo starting in 1998.
 
If there were 200 Bilbies in 1999 what would the population of the Bilbies be in 2004 ?
a)
859
 b)
451
c)
490
 d)
771

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
If Pn is the population of the Bilbies n years after the breeding program began then P1 = 200 in 1999 so we are looking for P6 in 2004 and
P2 = 334,P3 = 457.28,P4 = 570.7,P5 = 675.04,P6 = 771.04.