Exponential functions are functions defined by y = ab^x, where a represents the initial value, and b represents the rate
<h3>The equation of a city that has experienced a population growth</h3>
The initial population of the city is 10000, and the growth rate of the population is 4%.
So, the exponential equation is:
![y = 10000 * 1.04^x](https://tex.z-dn.net/?f=y%20%3D%2010000%20%2A%201.04%5Ex)
<h3>The equation of a city that has experienced a population decline</h3>
The initial population of the city is 12000, and the decay rate of the population is 3%.
So, the exponential equation is:
![y = 12000* 0.97^x](https://tex.z-dn.net/?f=y%20%3D%2012000%2A%200.97%5Ex)
<h3>The similarities in the equations</h3>
The similarity in both equations is that, they both represent exponential function.
<h3>The year the population of city A exceeds B</h3>
In (a) and (b), we have:
---- city A
--- city B
When city A exceeds city B, we have the following inequality
![10000 * 1.04^x > 12000 * 0.97^x](https://tex.z-dn.net/?f=10000%20%2A%201.04%5Ex%20%3E%2012000%20%2A%200.97%5Ex)
Divide both sides by 10000
![1.04^x > 1.2 * 0.97^x](https://tex.z-dn.net/?f=1.04%5Ex%20%3E%201.2%20%2A%200.97%5Ex)
Divide both sides by 0.97^x
![(\frac{1.04}{0.97})^x > 1.2](https://tex.z-dn.net/?f=%28%5Cfrac%7B1.04%7D%7B0.97%7D%29%5Ex%20%3E%201.2)
![1.07^x > 1.2](https://tex.z-dn.net/?f=1.07%5Ex%20%3E%201.2)
Take the natural logarithm of both sides
![\ln(1.07)^x > \ln(1.2)](https://tex.z-dn.net/?f=%5Cln%281.07%29%5Ex%20%3E%20%5Cln%281.2%29)
This gives
![x\ln(1.07) > \ln(1.2)](https://tex.z-dn.net/?f=x%5Cln%281.07%29%20%3E%20%5Cln%281.2%29)
Solve for x
![x > \frac{\ln(1.2)}{\ln(1.07)}](https://tex.z-dn.net/?f=x%20%3E%20%5Cfrac%7B%5Cln%281.2%29%7D%7B%5Cln%281.07%29%7D)
![x > 2.69](https://tex.z-dn.net/?f=x%20%3E%202.69)
This means that, the population of city A will exceed city B after 3 years
<h3>The year the population of city A will be at least twice of city B</h3>
In (a) and (b), we have:
---- city A
--- city B
When city A is at least twice city B, we have the following inequality
![10000 * 1.04^x \ge 2 * 12000 * 0.97^x](https://tex.z-dn.net/?f=10000%20%2A%201.04%5Ex%20%5Cge%202%20%2A%2012000%20%2A%200.97%5Ex)
![10000 * 1.04^x \ge 24000 * 0.97^x](https://tex.z-dn.net/?f=10000%20%2A%201.04%5Ex%20%5Cge%2024000%20%2A%200.97%5Ex)
Divide both sides by 10000
![1.04^x \ge 2.4 * 0.97^x](https://tex.z-dn.net/?f=1.04%5Ex%20%5Cge%202.4%20%2A%200.97%5Ex)
Divide both sides by 0.97^x
![(\frac{1.04}{ 0.97})^x \ge 2.4](https://tex.z-dn.net/?f=%28%5Cfrac%7B1.04%7D%7B%200.97%7D%29%5Ex%20%5Cge%202.4)
![1.07^x \ge 2.4](https://tex.z-dn.net/?f=1.07%5Ex%20%5Cge%202.4)
Take the natural logarithm of both sides
![\ln(1.07)^x \ge \ln(2.4)](https://tex.z-dn.net/?f=%5Cln%281.07%29%5Ex%20%5Cge%20%5Cln%282.4%29)
This gives
![x\ln(1.07) \ge \ln(2.4)](https://tex.z-dn.net/?f=x%5Cln%281.07%29%20%5Cge%20%5Cln%282.4%29)
Solve for x
![x\ge \frac{\ln(2.4)}{\ln(1.07) }](https://tex.z-dn.net/?f=x%5Cge%20%5Cfrac%7B%5Cln%282.4%29%7D%7B%5Cln%281.07%29%20%7D)
![x\ge 12.9](https://tex.z-dn.net/?f=x%5Cge%2012.9)
This means that, the population of city A will be at least twice city B after 13 years
Read more about exponential functions at:
brainly.com/question/11464095