Question 187051
" initial size 100 grows at the rate of 8% per year forever" ---> *[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]




What is the size of the population at the end of year 1?


*[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE P(1)=100\cdot\left(1.08\right)^{1}]


*[Tex \LARGE P(1)=100\cdot\left(1.08\right)]


*[Tex \LARGE P(1)=108]


Ans: 108 people at end of yr 1



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What is the size of the population at the end of year 2?


*[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE P(2)=100\cdot\left(1.08\right)^{2}]


*[Tex \LARGE P(1)=100\cdot\left(1.1664\right)]


*[Tex \LARGE P(1)=116.64]


Ans: 116 people at end of yr 2



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What is the size of the population at the end of year 3?


*[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE P(2)=100\cdot\left(1.08\right)^{3}]


*[Tex \LARGE P(1)=100\cdot\left(1.259712\right)]


*[Tex \LARGE P(1)=125.9712]


Ans: 125 people at end of yr 3



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What is the size of the population at the end of year n?


*[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE P(n)=100\cdot\left(1.08\right)^{n}]



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find the number of years x that it would take for our population to reach 200?


*[Tex \LARGE P(t)=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE 200=100\cdot\left(1.08\right)^{t}]


*[Tex \LARGE 2=\left(1.08\right)^{t}]


*[Tex \LARGE \log_{10}(2)=\log_{10}\left(\left(1.08\right)^{t}\right)]


*[Tex \LARGE \log_{10}(2)=t\cdot\log_{10}\left(1.08\right)]


*[Tex \LARGE \frac{\log_{10}(2)}{\log_{10}\left(1.08\right)}=t]


*[Tex \LARGE t=\frac{\log_{10}(2)}{\log_{10}\left(1.08\right)}]


*[Tex \LARGE t \approx 9.0064]



Ans: takes about 9 yrs for pop. to reach 200 people