Respuesta :
[tex]\begin{gathered} A)P(t)=30(1+0.088)^t\text{( in millions)} \\ B)P(20)=163.33\text{ Million people} \\ C)163.33\text{ Million people} \\ D)8.21years \end{gathered}[/tex]
Explanation
you can model growth by a constant percent increase with the following formula:
[tex]\begin{gathered} P(t)=A(1+r)^t \\ \text{where} \\ P(t)\text{ is the final amount} \\ A\text{ is }the\text{ initial amount} \\ r\text{ is the rate ( in decimal)} \\ t\text{ is the time ( years)} \end{gathered}[/tex]so
Step 1
make the model.
A)Let
[tex]\begin{gathered} P=70\text{ million} \\ \text{time}=\text{ 1990-180=10 years} \\ A=30\text{millones } \\ so \\ 70=30(1+r)^{10}\rightarrow equation \end{gathered}[/tex]now, we need to solve for r
[tex]\begin{gathered} 70=30(1+r)^{10}\rightarrow equation \\ \frac{70}{30}=(1+r)^{10} \\ (\frac{70}{30})^{\frac{1}{10}}=((1+r)^{10})^{\frac{1}{10}} \\ 1.088=1+r \\ \text{subtract 1 in both sides} \\ 1.088-1=1+r-1=0.0884229198 \\ r=0.0884229198 \\ or \\ r=8.8\text{ \%} \end{gathered}[/tex]c) now, we can complete the model
[tex]P(t)=30(1+0.088)^t[/tex]Step 2
Whar population do you predict for the year 2000?
Let
[tex]\begin{gathered} \text{time}=\text{ 2000-1980=20 years} \\ t=20 \\ P=30 \\ r=\text{0}.088 \end{gathered}[/tex]replace
[tex]\begin{gathered} P(20)=30(1+0.088)^{20} \\ P(20)=30(1+0.088)^{20} \\ P(20)=30(1+0.088)^{20} \\ P(20)=163.33\text{ Million people} \end{gathered}[/tex]Step 3
D) What is the doubling time?
Doubling time =____ years
to solve this , we need to find the time , when population is double than currently, so
[tex]\begin{gathered} \text{Double population= }2\cdot300\text{ milliones} \\ \text{Double population=}600\text{ Million} \end{gathered}[/tex]then, let
[tex]\begin{gathered} \text{time}=\text{ unknown= t} \\ A=30 \\ P=60 \\ r=0.088 \end{gathered}[/tex]replace
[tex]\begin{gathered} P(t)=30(1+0.088)^t \\ 60=30(1+0.088)^t \\ 60=30(1.088)^t \\ \frac{60}{30}=(1.088)^t \\ 2=(1.088)^t \\ \ln (2)=\ln (1.088)^t \\ \ln (2)=t\ln (1.088) \\ t=\frac{\ln 2}{\ln \text{ 1.088}} \\ t=8.2183 \end{gathered}[/tex]
therefore, the time is
D)8.21 years
I hope this helps you
