Question

I need help number 7

Answer 1

The population at the beginning of 1950 was 2600 thousand people.

Then it started increasing exponentially 23% every decade.

The general form of any exponential function is:

$f(x)=a(b)^x$

Where

a is the initial value

b is the growth/decay factor

x is the number of time periods

y is the final value after x time periods

a. To calculate the growth factor of an exponential function, you have to add the increase rate (expressed as a decimal value) to 1:

$\begin{gathered} b=1+r \\ b=1+\frac{23}{100} \\ b=1.23 \end{gathered}$

b. Considering the initial value a= 2600 thousand people and the growth factor b=1.23, you can express the exponential function in terms of the number of decades, d, as follows:

$f(d)=2600(1.23)^d$

c. Considering that the time unit is measured in decades, i.e d=1 represents 10 years

To determine the corresponding value of the variable d for 1 year, you have to divide 1 by 10

$1\text{year/10years d}=\frac{1}{10}=0.1$

Calculate the growth factor powered by 0.1:

$\begin{gathered} b_{1year}=(1.23)^{0.1} \\ b_{1year}=1.0209\approx1.02 \end{gathered}$

d. Use the factor calculated in item c

$g(t)=2600(1.0209)^t$