Exponential Growth & Decay: Calculus 1 Project by Ignacio Rendon
Overview
This Project has been completed as part of a standard 10 weeks Calculus 1 asynchronous online course with optional WebEx sessions during Summer 2021 Semester at MassBay Community College, Wellesley Hills, MA.
Exponential Growth & Decay
INTRO
Exponential growth and decay are formulas that are very commonly used in the real world to predict trends and changes of something. In simple terms Exponential growth is the increases of something at contstant rate proportional to it's size, the same applies for decay except the value would be decreasing at a constant rate. There are two key words that really define exponential growth, those are proportional and constant. Let's take a look at Exponential growth first.
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BODY
Exponential Growth:
When we look up Exponential Growth on the dictionary it defines it as "Growth of a system in which the amount being added to the system is proportional to the amount already present: the bigger the system is, the greater the increase."
Although there is no single person credited to the discovery or creation of Exponential growth and decay, we can still give credit to the English economist and scholar Thomas Malthus who came up with the formula while trying to prove that population growth was related to the economy in the late 1700's.
$$P(t)=P_0e^{rt}$$
\(P_0\) = initial Population
\(r\) = population growth rate
\(t\) = time
Even though this formula is still valid, we are more used to seeing this formula for calculating Exponential Growth:
$$f(x)=a(1+r)^t$$
There are also other formulas used to calculate exponential growth shown below:
Real World Uses of Exponential Growth:
One of the most common uses of Exponential Growth is to calculate population growth, but there are still many other important uses such as:
Compound Interest - Compound Interest at a constant rate provides exponential growth to capital
Biology - Studying the growth at which micro-organisms reproduce in a suitable evniorment. This can help with calculating when food will start going bad to give products expiration rates.
Wildfires - With the relation to the enviorment aroun it, we can calculate the exponential rate at which the fire will continue to spread
Diseases - Cancer, cancer cells spread exponentially if not treated. Another example would be major pandemics, like covid 19.
Invasive species - These can be any living thing that is not native to a location and causes harm to the enviormant. Some examples would be the Water Hyancith weed or the Japonese Honeysuckle
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Exponential Decay:
"In mathematics, exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time."
$$y=a(1-r)^x$$
\(y\)= final value
\(a\)= starting value
\(r\)= decay rate
\(x\)= time
Same as Exp. growth there are other formulas we can use to calculate Exponential Decay:
Real World Uses of Exponential Decay:
Compound formulas: Same as growth but this time it is measuring the loss of capital
Half-Life of Radiactive substances
Carbon Dating
Depreciation - measuring the loss of value of something over time (car, phone ect)
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Half-Life and Radioactive Decay
In Chemistry, Half-Life is an important topic that gets brought up a lot, especially when dealing when dealing with radioactive substances. What is half-life? "Half-Life is the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay."
Formulas commonly used to find half-life probelms:
$$N=N_0\Big( \dfrac{1}{2}\Big)^{\frac{t}{h}}$$
\(N_0\) = Initial amount
\(N\) = Final amount
\(t\)= Time
\(h\)= time for 1/2 of sample to decay
Ex. Half life of Carbon-14 is 5,730 years. How much of a 10.0mg sample will we have after 4,500 years have elapsed?
\(N=(10.0)\Big(\dfrac{1}{2} \Big)^{\frac{4500}{5730}}\\ N=(10.0)\Big(\dfrac{1}{2} \Big)^{0.7853}\\ N=(10.0)(0.5802)\\ N=5.802~mg \)
SOURCES:
Human Population Growth | Biology for Majors II