Question

there's 2 parts of the question.Also , one of the last tutors did multiple questions and helped me with about most of my homework... can you help me with other questions than this too , thank you anyway though.

Answer 1

Part A:

In 2000, the population was 6.26 million.

In 2015, the population was 7 million.

The general form of the exponential function is given by

$A=A_0e^{k\cdot t}$

Where A0 is the initial population (6.26 million), k is the growth rate, and t is the number of years after 2000.

Let us first find the growth rate.

We are given that In 2015, the population was 7 million.

$\begin{gathered} A=A_0e^{k\cdot t} \\ 7=6.26e^{k\cdot15} \\ \frac{7}{6.26}=e^{k\cdot15} \\ \ln (\frac{7}{6.26})=\ln (e^{k\cdot15}) \\ 0.1117=k\cdot15 \\ k=\frac{0.1117}{15} \\ k=0.0074 \\ k=0.01 \end{gathered}$

So, the growth rate is 0.01 (rounded to 2 decimal places)

Therefore, the exponential growth model is

$A=6.26e^{0.01t}$

Part B:

We need to find the year for which the population will be 12 million.

Let us substitute A = 12 into the exponential growth function and solve for t

$\begin{gathered} A=6.26e^{0.01t} \\ 12=6.26e^{0.01t} \\ \frac{12}{6.26}=e^{0.01t} \\ \ln (\frac{12}{6.26})=\ln (e^{0.01t}) \\ 0.6507=0.01t \\ \frac{0.6507}{0.01}=t \\ 65=t \\ t=65\: \text{years} \end{gathered}$

Therefore, the country's population will be 12 million in the year 2065