Answer:
a) r = 0.79%
b)
[tex]P_t=5.17(1.0079)^t[/tex]
c) 6.6 million people
d) 2046
Explanation:
We'll use the below formula for exponential growth;
[tex]P_t=a(1+r)^t[/tex]
where a = initial amount
r = growth rate
t = number of time intervals
a) From the question, we have that
a = 5.17 million
P(t)= 6.05 million
t = 20 years
Let's go ahead and substitute these values into our formula, and solve for r as shown below;
[tex]\begin{gathered} 6.05=5.17(1+r)^{20} \\ \frac{6.05}{5.17}=(1+r)^{20} \\ (1+r)=\sqrt[20]{\frac{6.05}{5.17}} \\ r=\sqrt[20]{\frac{6.05}{5.17}}-1 \\ r=0.00789 \\ r=0.79\text{\%} \end{gathered}[/tex]
b) The exponential model can be written as shown below;
[tex]\begin{gathered} P_t=5.17(1+0.0079)^t \\ P_t=5.17(1.0079)^t \end{gathered}[/tex]
c) When t = 31 years, let's go ahead and find P as shown below;
[tex]\begin{gathered} P_t=5.17(1.0079)^{31} \\ P_t=6.6\text{ million people} \end{gathered}[/tex]
d) When P = 7.5 million, let's go ahead and solve for t as shown below;
[tex]\begin{gathered} 7.5=5.17(1.0079)^t \\ 1.45=(1.0079)^t \\ \log 1.45=\log (1.0079)^t \\ \log 1.45=t\times\log (1.0079) \\ t=\frac{\log 1.45}{\log (1.0079} \\ t=47.2\text{years} \\ \end{gathered}[/tex]
So to get the particular year all we need to do is add 47 years to the initial year. That will us 1999 + 47 = 2046
