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# 18.8: Growth Models

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1.

1. $$P_{0}=20 . P_{n}=P_{n-1}+5$$
2. $$P_{n}=20+5 n$$

3.

1. $$P_{1}=P_{0}+15=40+15=55 . P_{2}=55+15=70$$
2. $$P_{n}=40+15 n$$
3. $$P_{10}=40+15(10)=190$$ thousand dollars
4. $$40+15 n=100$$ when $$n=4$$ years.

5. Grew 64 in 8 weeks: 8 per week

1. $$P_{n}=3+8 n$$
2. $$187=3+8 n . n=23$$ weeks

7.

1. $$P_{0}=200$$ (thousand), $$P_{n}=(1+.09) P_{n-1}$$ where $$n$$ is years after 2000
2. $$P_{n}=200(1.09)^{n}$$
3. $$P_{16}=200(1.09)^{16}=794.061(\text { thousand })=794,061$$
4. $$200(1.09)^{n}=400 . \quad n=\log (2) / \log (1.09)=8.043 .$$ In 2008

9. Let $$n=0$$ be $$1983 . \quad P_{n}=1700(2.9)^{n} . \quad 2005$$ is $$n=22 . \quad P_{22}=1700(2.9)^{22}=25,304,914,552,324$$ people. Clearly not realistic, but mathematically accurate.

11. If n is in hours, better to start with the explicit form. $$P_{0}=300 . P_{4}=500=300(1+r)^{4}$$

$$500 / 300=(1+r)^{4} . \quad 1+r=1.136 . \quad r=0.136$$

1. $$P_{0}=300 . \quad P_{n}=(1.136) P_{n-1}$$
2. $$P_{n}=300(1.136)^{n}$$
3. $$P_{24}=300(1.136)^{24}=6400$$ bacteria
4. $$300(1.136)^{n}=900 . n=\log (3) / \log (1.136)=$$ about 8.62 hours

13.

1. $$P_{0}=100 \quad P_{n}=P_{n-1}+0.70\left(1-P_{n-1} / 2000\right) P_{n-1}$$
2. $$P_{1}=100+0.70(1-100 / 2000)(100)=166.5$$
3. $$P_{2}=166.5+0.70(1-166.5 / 2000)(166.5)=273.3$$

15. To find the growth rate, suppose $$n=0$$ was $$1968 .$$ Then $$P_{0}$$ would be 1.60 and $$P_{8}=2.30=1.60(1+r)^{8}, r=0.0464 .$$ since we want $$n=0$$ to correspond to 1960 , then we don't know $$P_{0}$$, but $$P_{8}$$ would $$1.60=P_{0}(1.0464)^{8}$$. $$P_{0}=1.113$$

1. $$P_{n}=1.113(1.0464)^{n}$$
2. $$P_{0}=\ 1.113,$$ or about $$\ 1.11$$
3. 1996 would be $$n=36 . \quad P_{36}=1.113(1.0464)^{36}=\ 5.697 .$$ Actual is slightly lower.

17. The population in the town was 4000 in 2005, and is growing by 4% per year.

18.8: Growth Models is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.