Some Qs in Exponential Growth: Compute population differences, find growth rate, ...

[FelipeAlfonso]

Hey y'all! I just joined so hi hehe.

I'm horrible at articulating and organizing the ideas of my question. I'm just gonna post some of the questions in our task and look for the gist. I wonder how ya'll would solve this

Given that the population in:
1980: 5,925,884
1990: 7,948,392
2000: 9,932,560
2010: 11,855,975

I'm already done. I'm just kind of not confident with my answer since when I reverse solve it, the estimation is not exactly as it should be but somewhat near the value.
Anyways, thanks in advance!

 

[mmm4444bot]

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[JeffM]

Hey y'all! I just joined so hi hehe.

I'm horrible at articulating and organizing the ideas of my question. I'm just gonna post some of the questions in our task and look for the gist. I wonder how ya'll would solve this

Given that the population in:
1980: 5,925,884
1990: 7,948,392
2000: 9,932,560
2010: 11,855,975
View attachment 10512
I'm already done. I'm just kind of not confident with my answer since when I reverse solve it, the estimation is not exactly as it should be but somewhat near the value.
Anyways, thanks in advance!

First, please read this.

https://www.freemathhelp.com/forum/threads/109845-Guidelines-Summary?p=422890&viewfull=1#post422890

Second, if you give us your answers, we shall tell you whether they are correct or not. Of course, if all we see is your answer, we cannot tell you where you went wrong if the answer is incorrect. Alternatively, you can show your answer along your work (even if you suspect it to be wrong), and then we can either confirm your answer or help you correct your mistake.

Third, we prefer if you post one question per thread.

 

[FelipeAlfonso]

Oh. I'm really sorry. I'm in Grade 11 studying Exponential Growth and Decay.
Here's how my work goes:

Given that the population in:
1980: 5,925,884
1990: 7,948,392
2000: 9,932,560
2010: 11,855,975

1. Compute the difference between the population size of the region between 1980 and 1990, 1990 and 2000,
2000 and 2010.

(I subtract)
Between 1990 and 1980: 2,022,508
Between 2000 and 1990: 1,984,168
Between 2010 and 2000: 1,923,415


2. Determine the rate of growth of the population every ten years by dividing each difference by the population
of the previous year. Multiply the result by 100% and express the rate of growth up to 2 decimal places.

2,022,508/7,948,392= 0.25445 x 100 = 25.45%
1,984,168/9,932,560= 0.19976 x 100 = 19.98%
1,923,415/11,855,975= 0.16223 x 100 = 16.22%


3. Find the average (arithmetic mean) of the three rates then divide it by 10. Result serves as the annual relative
rate of growth of the region's population. Express this up to 2 decimal places.

0.2545 + 0.1998 + 0.1622 / 3 = 0.2055 or 20.55%/10 = 2.06%


4. Prepare the population model of the region by following the exponential growth model given as:
n(t) = n₀e^rt
where n(t) = population at time t
n₀ = initial size of the population
r = relative rate of growth (expressed as a proportion of the population)
t = time
Note that r is the result obtained in Step number 2, t is the number of years after 1980, and n0 is the
population in 1980. Convert the annual rate of growth (r) obtained in Step 2 up to 4 decimal places (i.e.
Step 2: r = 1:75%; Step 4: r = 0:0175)

Uhh I don't really get this/how should I model it

5. Using the model in Step number 5, estimate the population of the region in:
a. 2018
2018-1980=38, n(38)=5925884e^0.0206(38)
12,963,398.34
b. 2019
2019-1980=39, n(39)=5925884e^0.0206(39)
13,233,213.91
c. 2020
2020-1980=40, n(40)=5925884e^0.0206(40)
13,508,645.32
d. 2030
2030-1980=50, n(50)=5925884e^0.0206(50)
16,598,791.21
e. 2040
2040-1980=60, n(60)=5925884e^0.0206(60)
20,395,817.89


6. Using your knowledge of logarithms, estimate the year in which the population will reach
a. 30 million
30000000=5925884e^0.0206t, 5.062535817=e^0.0206t, ln(5.062535817)=0.0206t*ln(e), ln(5.06253817)/0.0206=t,
t=78.73 (Exact Value: 78.7314324), 1980+78.73= 2058.73
b. 50 million
50000000=5925884e^0.0206t, 8.437559696=e^0.0206t​, ln(8.437559696)=0.0206t*ln(e), ln(8.437559696)/0.0206=t,
t=103.53 (Exact Value: 103.5287928), 1980+103.53= 2083.53

Those are my answers. I'm not really sure if they're totally correct. If I input only 2 decimal places on the calculator to prove that the equal value is true, it won't give me the EXACT answer but near to very near the value (can be rounded off) so I guess that's okay? E.g., 30000000=5925884e^{0.0206(78.73)} is only equals to 29,999,114.79. But if I put the exact value of T (78.7314324), the answer is exactly 30,000,000. In the annual relative growth, if I put the exact value vs the rounded off one, the value will not be as exact but still near. I tried to reverse solve it, using the formula that I've made and concluded, the population of 2010 is 11,855,975, so 2010-1980=30, 5925884e^0.0206(30)=11855875, but 10,993,782 is not equal to 11,855,975. Did I do something, please help I'm really confused on what would be the basis of the values that I should use, should it be the exact whole value then round it off to 2 decimal places or round it off first as instructed by the problem?

 

[mmm4444bot]

… I'm really sorry …

Okay, but I think you missed the following. Your three duplicated posts just now have been deleted. There's no need to post duplicated messages.

NOTE 2: The first few posts from each new member will not appear on the boards until after they have been checked by a moderator. Please see this notice.

 

[Steven G]

2. Determine the rate of growth of the population every ten years by dividing each difference by the population
of the previous year. Multiply the result by 100% and express the rate of growth up to 2 decimal places.

2,022,508/7,948,392= 0.25445 x 100 = 25.45%
1,984,168/9,932,560= 0.19976 x 100 = 19.98%
1,923,415/11,855,975= 0.16223 x 100 = 16.22%

I am a very big believer that equal signs MUST be valid.

For example you wrote 2,022,508/7,948,392= 0.25445 x 100 = 25.45% even though 2,022,508/7,948,392\(\displaystyle \neq\) 0.25445 x 100 and 0.16223 x 100 \(\displaystyle \neq\) 16.22%.

So let's clean this up. If you want to change a number to a percentage you DO NOT multiply by 100, regardless of who told you this! Whenever you want to change the way something looks, you multiply it by 1. If you want to change something so it has a % sign you multiply by 1!!! How should you write the 1? Simple, 100% = 1 AND it has a % sign.

Instead of 2,022,508/7,948,392= 0.25445 x 100 = 25.45% you should write 2,022,508/7,948,392= 0.25445 x 100% = 25.45%

Same for the other two equations.

 

[JeffM]

Oh. I'm really sorry. I'm in Grade 11 studying Exponential Growth and Decay.
Here's how my work goes:

Given that the population in:
1980: 5,925,884
1990: 7,948,392
2000: 9,932,560
2010: 11,855,975

1. Compute the difference between the population size of the region between 1980 and 1990, 1990 and 2000,
2000 and 2010.

(I subtract)
Between 1990 and 1980: 2,022,508
Between 2000 and 1990: 1,984,168
Between 2010 and 2000: 1,923,415

THIS IS CORRECT

2. Determine the rate of growth of the population every ten years by dividing each difference by the population
of the previous year. Multiply the result by 100% and express the rate of growth up to 2 decimal places.

2,022,508/7,948,392= 0.25445 x 100 = 25.45%
1,984,168/9,932,560= 0.19976 x 100 = 19.98%
1,923,415/11,855,975= 0.16223 x 100 = 16.22%

THIS IS IN ERROR. Growth between 1980 and 1990 was 2,022,508, which was the actual growth from 1980, the EARLIER year. You should be dividing by the population for 1980.

3. Find the average (arithmetic mean) of the three rates then divide it by 10. Result serves as the annual relative
rate of growth of the region's population. Express this up to 2 decimal places.

0.2545 + 0.1998 + 0.1622 / 3 = 0.2055 or 20.55%/10 = 2.06%

4. Prepare the population model of the region by following the exponential growth model given as:
n(t) = n₀e^rt
where n(t) = population at time t
n₀ = initial size of the population
r = relative rate of growth (expressed as a proportion of the population)
t = time
Note that r is the result obtained in Step number 2, t is the number of years after 1980, and n0 is the
population in 1980. Convert the annual rate of growth (r) obtained in Step 2 up to 4 decimal places (i.e.
Step 2: r = 1:75%; Step 4: r = 0:0175)

Uhh I don't really get this/how should I model it

5. Using the model in Step number 5, estimate the population of the region in:
a. 2018
2018-1980=38, n(38)=5925884e^0.0206(38)
12,963,398.34
b. 2019
2019-1980=39, n(39)=5925884e^0.0206(39)
13,233,213.91
c. 2020
2020-1980=40, n(40)=5925884e^0.0206(40)
13,508,645.32
d. 2030
2030-1980=50, n(50)=5925884e^0.0206(50)
16,598,791.21
e. 2040
2040-1980=60, n(60)=5925884e^0.0206(60)
20,395,817.89


6. Using your knowledge of logarithms, estimate the year in which the population will reach
a. 30 million
30000000=5925884e^0.0206t, 5.062535817=e^0.0206t, ln(5.062535817)=0.0206t*ln(e), ln(5.06253817)/0.0206=t,
t=78.73 (Exact Value: 78.7314324), 1980+78.73= 2058.73
b. 50 million
50000000=5925884e^0.0206t, 8.437559696=e^0.0206t​, ln(8.437559696)=0.0206t*ln(e), ln(8.437559696)/0.0206=t,
t=103.53 (Exact Value: 103.5287928), 1980+103.53= 2083.53

Those are my answers. I'm not really sure if they're totally correct. If I input only 2 decimal places on the calculator to prove that the equal value is true, it won't give me the EXACT answer but near to very near the value (can be rounded off) so I guess that's okay? E.g., 30000000=5925884e^{0.0206(78.73)} is only equals to 29,999,114.79. But if I put the exact value of T (78.7314324), the answer is exactly 30,000,000. In the annual relative growth, if I put the exact value vs the rounded off one, the value will not be as exact but still near. I tried to reverse solve it, using the formula that I've made and concluded, the population of 2010 is 11,855,975, so 2010-1980=30, 5925884e^0.0206(30)=11855875, but 10,993,782 is not equal to 11,855,975. Did I do something, please help I'm really confused on what would be the basis of the values that I should use, should it be the exact whole value then round it off to 2 decimal places or round it off first as instructed by the problem?

Thank you for showing your work in such detail.

You made a mistake in step 2. You must divide by the population for the earlier year. You are taking the change from the earlier period to the later period and finding out what that is relative to the earlier period. In your instructions, they say "previous," but it might have been clearer to say "earlier."

So let's start by having you fix that mistake.

You must follow your instructions, but I would prefer it had they asked you to use the geometric mean rather than the arithmetic mean. The difference, however, is small. I calculate the arithmetic mean as about 26.15% whereas I calculate the geometric mean at about 26.01%. When you calculate the annual percentage rate of increase to two decimals, you get 2.61% from the arithmetic mean and 2.60% from the geometric mean. All the rest of your steps are wrong because you used 0.0206 rather than 0.0261.

I am not going to check your arithmetic in step 5 because you used the wrong r from step 2. In step 6, your work is a bit hard to follow, but I think you have the right idea (with the wrong numbers).