Question

# Suppose that a population that is growing exponentially increases from 800,000 people in 2007 to 1,000,000 people in 2010. Without showing the details

Exponential models
Suppose that a population that is growing exponentially increases from 800,000 people in 2007 to 1,000,000 people in 2010. Without showing the details, describe how to obtain the exponential growth function that models the data.

2021-05-15
Exponential Model : t : time , k : growth rate Given Values
Simplifying to find the value of k.
Substitute the given values to find exact value of k.
Substitute the value of k in the model which now varies only with t.
$$\displaystyle{A}={A}{0}{e}^{{k}}{t}$$
A0=800000
A=1000000
t=2
$$\displaystyle\frac{{A}}{{A}}{0}={e}^{{k}}{t}$$
$$\displaystyle{k}{t}={\ln{{\left(\frac{{A}}{{A}}{0}\right)}}}$$
$$\displaystyle{k}={\left(\frac{{1}}{{t}}\right)}{\ln{{\left(\frac{{A}}{{A}}{0}\right)}}}$$
$$\displaystyle{k}={\left(\frac{{1}}{{2}}\right)}{\ln{{\left(\frac{{1000000}}{{800000}}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\ln{{1.25}}}=\frac{{.22324}}{{2}}={.112}$$
$$\displaystyle{A}={800000}{e}^{{.112}}{t}$$