Modu egnit Growth Fadtors 4: Investig Partial and n-unit Growt yearys ( Urban pl

Question

Modu egnit Growth Fadtors 4: Investig Partial and n-unit Growt yearys ( Urban planners noticed that the population of a certain county doubled over 20 a. What was the population of the county at the end of 2010? b. If the growth pattern people a 3 2010), following apattern ofexponential growth. The population ofthe county was 756000990 the end of 1990 were to continue for another 20 years, what is the projected population end of 2030? c. Which quantities are changing in this situation? Which quantities are not changing? d. If we assume that this population growth will continue into the far future, complete the table. current times as large as the population at the end of 1990 Number of years since 1990 Population 20 40 60 80 e. Define a function fthat expresses the population of the county, P after a number of 20-year periods, n, that have elapsed since the end of 1990. 4Since 20-year changes in time are quite long, we might want to know how much the population changes over shorter periods of time. a. Given that the population grew exponentially, which of the following two statements (i. and ii.) best describes what happens every 10 years? Justify your choice. iEvery 10 years, the population increases by a constant amount. That is, a constant amount is added to the population at the end of 1990 to get the population at the end of 2000, and the sam e amount is added to get the population at the end of 2010. ii. Every 10 years, the population increases by a constant factor. That is, this constant factor is multiplied by the population at the end of 1990 to get the population at the end of 2000, and this con the end of 2010. stant factor is multiplied by the population at the end of 2000 to get the population at b. Given that the population increases exponentially and doubles over i. What is the 10-year growth factor? ii. What is the I-year growth factor? Pathways Precalelus

Explanation / Answer

3.a) Since as it is given in the question that the population is doubling in 20 years. So, the population at the end of 2010 is :-

=> 2* population in 1990

=> 2*756000= 1512000

b) Population at the end of 2030:

2*Population at the end of year 2010

2*1512000=3024000

c) Quantity which is changing in the solution: Population

Quantity which is not changing in the solution: Rate of change of population.

d)

No of years since

1990

The current population is ------ times as largge

as the population at the end of year 1990

e) The required function to find the population at the end of 'n' years and which is increasing with the rate of 'r' exponentially is given by:

P(t)=Aert   

where P(t) is the population after the end of 't' years duration,

A is the population at the end of first year

r is the rate by which the population is increasing and

t is the duration in years.

4.a.) The correct answer is option (i). This is because the population is changing by the rate. And to get the population we need the rate to multiply with the initial population. So, adding will not give the correct result.

b) Population is given by :

  P(t)=Aert ,r is rate of change of population and t is the duration

since P(t)=2A

then

2A=Aer20 ,where r=rate of change of population

or, 2=er20

or taking loge on boht the sides, we get:

loge2 = loge er20

or,loge2 = r20logee

or , loge2 =r20, since logee =1

or ,0.3012=r20, loge2=0.3012

or, r=0.3012/20

or, r ,20 year growth factor is : 0.01505

therefore growth factor fo 10 years : 0.01505/2=0.007525

(ii) Growth factor for 1 year = growth factor for 10 years/10

= 0.007525/10

= 0.0007525

2) given values of 'y':

y1=38

y2=42

y3 =46

y4  =50

now the change is given by:

((final value - intial value )/(initial value))*100

so we get:

((42-38)/38)*100= (4/38)*100 =10.52

((46-42)/42)*100=(4/42)*100 = 9.52

((50-46)/46)*100=(4/46)*100 = 8.69

So from the above we can see that the difference change is not same in all the case.

To have exponential growth, we should have same change differences.

Hence, ordered pairs in the bold are not accurate ordered pairs.

No of years since

1990

Population

The current population is ------ times as largge

as the population at the end of year 1990

0 756000 1 20 1512000 2 40 3024000 4 60 6048000 8 80 12096000 16

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