Answer to Question #159141 in Algebra for Jan

Question

*CREATE a real-life exponential growth OR exponential decay word problem using real numbers and formulas

*determine where we see exponential growth and exponential decay in real life. For example, an exponential growth function can model the number of COVID-19 cases with respect to time. Reminder: DO NOT solve your problem. You simply need to create the question. 

Answer 1

Examples of exponential decay:

1. radioactive decay
2. population decrease

Examples of exponential growth:

1.Spoilage of Food

2.Human Population

3.Compound Interest

4.Pandemics

Exponential Growth is a mathematical function that can be used in several situations. The formula tells us the number of cases at a certain moment in time, in the case of Coronavirus, this is the number of infected people.

In other use cases of exponential growth, this number could be the size of an animal population or the value on your bank account.

The formula of Exponential Growth

Exponential Growth is characterized by the following formula:

$x(t) = x'*(b^t)$

The Exponential Growth function In which:

• x(t) is the number of cases at any given time t
• x' is the number of cases at the beginning, also called initial value
• b is the number of people infected by each sick person, the growth factor

A simple case of Exponential Growth: base 2

To make this more clear, I will make a hypothetical case in which:

• we start with an initial value of 1 infected person, so x'= 1
• each sick person infects 2 other people, so the growth rate b = 2
• we will inspect the development of the epidemic from time 0 to time 14

We first need to plug the values for a and b in the formula to obtain the formula for our specific epidemic:

$x=1*(2)^t$

Then we can use this formula to compute the value of y for each value of t from 0 to 14. When we do this, we obtain the following numbers of Infected people at every time step, as seen in the below table. This shows that starting from 1 person and with a growth factor of 2 per person, we obtain more than 16000 cases after 14 days.

time 0 day --- infected persons is 1.

time 1 day --- infected persons is 2

time 2 day --- infected persons is 4

time 3 day --- infected persons is 8

time 4 day --- infected persons is 16

time 5 day --- infected persons is 32

time 6 day --- infected persons is 64

and so on till day 14

time 14 day --- infected persons is 16384.

this shows the gradually increase in infected persons in case of exponential increase in positive cases.

Any graph that looks like the above (big on the left and crawling along the x-axis on the right) displays exponential decay, rather than exponential growth. For a graph to display exponential decay, either the exponent is "negative" or else the base is between 0 and 1.

It's exponential growth when the base of our exponential is bigger than 1, which means those numbers get bigger. It's exponential decay when the base of our exponential is in between 1 and 0 and those numbers get smaller.