Creating a Model for Population Growth

[NoGoodAtMath]

The population of Canada in 2010 was approximately 34 million with an annual growth rate of 0.804%. At this rate, the population P(t) (in millions) can be approximated by P(t)= 34(1.00804)^t, where t is the time in years since 2010.
a. Is the graph of P an increasing or decreasing exponential function?

Answer(2010)= 34(1.00804)^2010 approx 332,497,148 so it's increasing.

b. Evaluate P(0) and interpret its meaning in the context of this problem.

Answer: P(0)= 34(1.00804)^0= 34. My question is the interpretation, is 0 supposed to be the year 2000? I'm confused.

c. Evaluate P(15) and interpret its meaning in the context of this problem. round the population value to the nearest million.

Answer: P(15) = 34(1.00804)^15 is approx 38. My question is like the one above. Would the answer be something like the population in Canada in 2015 will be approximately 38 million if this trend continues?

 

[stapel]

The population of Canada in 2010 was approximately 34 million with an annual growth rate of 0.804%. At this rate, the population P(t) (in millions) can be approximated by P(t)= 34(1.00804)^t, where t is the time in years since 2010.
a. Is the graph of P an increasing or decreasing exponential function?

Answer(2010)= 34(1.00804)^2010 approx 332,497,148 so it's increasing.

Yes.

Quicker: The rate constant is greater than 1, so positive whole-number powers of t will make the base bigger. Thus, the values get bigger as t gets bigger. Thus, the function is increasing.

b. Evaluate P(0) and interpret its meaning in the context of this problem.

Answer: P(0)= 34(1.00804)^0= 34. My question is the interpretation, is 0 supposed to be the year 2000? I'm confused.

If the population in 2010 was 34 millions, and P(0) = 34 in millions, then how are you getting that 0 represents ten years earlier?

c. Evaluate P(15) and interpret its meaning in the context of this problem. round the population value to the nearest million.

Answer: P(15) = 34(1.00804)^15 is approx 38. My question is like the one above. Would the answer be something like the population in Canada in 2015 will be approximately 38 million if this trend continues?

On what basis are you obtaining "t = 0" means "2000" and "t = 15" means "2015"? Does this match what you've been given? Does this fit in the given equation?

 

[Ishuda]

The population of Canada in 2010 was approximately 34 million with an annual growth rate of 0.804%. At this rate, the population P(t) (in millions) can be approximated by P(t)= 34(1.00804)^t, where t is the time in years since 2010.
a. Is the graph of P an increasing or decreasing exponential function?

Answer(2010)= 34(1.00804)^2010 approx 332,497,148 so it's increasing.

b. Evaluate P(0) and interpret its meaning in the context of this problem.

Answer: P(0)= 34(1.00804)^0= 34. My question is the interpretation, is 0 supposed to be the year 2000? I'm confused.

c. Evaluate P(15) and interpret its meaning in the context of this problem. round the population value to the nearest million.

Answer: P(15) = 34(1.00804)^15 is approx 38. My question is like the one above. Would the answer be something like the population in Canada in 2015 will be approximately 38 million if this trend continues?

(a) That works in the case of a simple exponential but, in general, just because the value of a function is larger for a larger argument, that does not mean the function is increasing.
(b) ..."My question is the interpretation, is 0 supposed to be the year 2000? I'm confused." The variable t is the time in years since 2010. So if t=0 what year is it?
(c) If t=15 what year is it?

 

[NoGoodAtMath]

(a) That works in the case of a simple exponential but, in general, just because the value of a function is larger for a larger argument, that does not mean the function is increasing.
(b) ..."My question is the interpretation, is 0 supposed to be the year 2000? I'm confused." The variable t is the time in years since 2010. So if t=0 what year is it?
(c) If t=15 what year is it?

If t=0 I think it's 2010 and the population barely increased.
If t=15 I think it's 2015 and since then the population has increased significantly and is approximately 38 million.

It looks like 2010 and 2015 population is being compared.

 

[stapel]

If t=0 I think it's 2010 and the population barely increased.
If t=15 I think it's 2015 and since then the population has increased significantly and is approximately 38 million.

The variable "t" is defined as indicating the number of years after 2010. How are you getting that the 2010 population has changed "since" 2010? How are you getting that 2015 is 15 years after 2010?

 

[Ishuda]

If t=0 I think it's 2010 and the population barely increased.

Yes. If t=0 then it is 0 years from 2010 and the population is 'approximately 34 million'

If t=15 I think it's 2015 and since then the population has increased significantly and is approximately 38 million.

If t=15 then it is 15 years from 2010. So what year is it? You are correct in that the population is approximately 38 million.

It looks like 2010 and 2015 population is being compared.

Yes, for the specific example the population has grown by about 4 million in 15 years. In general, I would say that the yearly population was being compared to that of 2010.

 

[NoGoodAtMath]

Yes. If t=0 then it is 0 years from 2010 and the population is 'approximately 34 million'


If t=15 then it is 15 years from 2010. So what year is it? You are correct in that the population is approximately 38 million.


Yes, for the specific example the population has grown by about 4 million in 15 years. In general, I would say that the yearly population was being compared to that of 2010.

So, t= 2025 By 2025, the population growth is approximately 34 million.

 

[Steven G]

The population of Canada in 2010 was approximately 34 million with an annual growth rate of 0.804%. At this rate, the population P(t) (in millions) can be approximated by P(t)= 34(1.00804)^t, where t is the time in years since 2010.
a. Is the graph of P an increasing or decreasing exponential function?

Answer(2010)= 34(1.00804)^2010 approx 332,497,148 so it's increasing. No! Some functions increase and then decrease or stay constant for a while. It is true however that A^t is an increasing function if A>0 ( and decreasing function if 0<t<1)

b. Evaluate P(0) and interpret its meaning in the context of this problem.

Answer: P(0)= 34(1.00804)^0= 34. My question is the interpretation, is 0 supposed to be the year 2000? I'm confused. where t is the time in years since 2010, Since 2010 is 0 years from 2010 it follows that when t=0, we are talking about 2010

c. Evaluate P(15) and interpret its meaning in the context of this problem. round the population value to the nearest million.

Answer: P(15) = 34(1.00804)^15 is approx 38. My question is like the one above. Would the answer be something like the population in Canada in 2015 will be approximately 38 million if this trend continues?

See red above