# Exponential Growth And Decay

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```					             Exponential Growth And Decay
Exponential Growth And Decay

Today we will discuss about Exponential Growth and Decay. First we will pay attention
towardsexponential growth.

Exponential growth simply represents the growth of any value of the mathematical function
proportional to the function’s present value. When the system is distributed or discreet then
the intervals formed are known as geometric growth.

The general formula of the exponential growth of any variable ‘q’ at the growth rate ‘R’ and
time interval‘t’ comes in discrete intervals

Q(t) = q0(1+r)^t

Here the q is the variable. R is the rate that represents that next time the rate will be r times.
Suppose the rate is 6 percent. Then the current rate of the variable ‘q’ is 0.06 and next time
the rate will be 1.06 times the previous time.

The basic formula of the exponential growth is:

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q(t) = c*n^(t/r)

Where the ‘c’ is the constant value and the initial value of ‘q’, that is q(0)   =   c

Here ‘n’ is the positive growth factor and‘t’ is the time required to for ‘q’ to increase by a factor
of ’n’

q(t=R) = q(t)* n

If R>0 and n>1 then ‘q’ has exponential growth. But if R<0 and n>1 or R>0 and 0< n <1,
then it called as exponential decay.

Let us take one example.

Question 1. A virus doubles in every ten minutes, starting out with one, how many viruses will
be present after one hour.

Answer 1. Here in the above example c= 1 and n = 10 and rate is 10 min

Q(t) = c*n^(t/r)

Q(1 hour)=     1*2^6 = 64

So after one hour there will be 64 viruses.

Applications of exponential growth:

1.     Certain microorganisms reproduce in exponential form. They split into its daughter cells.

2.     Every epidemic or virus always spread exponentially.

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3.     Nuclear chain reaction is an example of exponential growth.

4.     Heat transfer is also done exponentially.

5.     Economic growth of a country is determined by the exponentially analyzing it.

6.     In fact the Moore’s law is also based on exponential order.

Now let us talk about exponential decay. Any quantity which decreases at a rate proportional
to its value, is known as exponential decay.

This can be represented as

dq / dt =    -λq

Where ‘q’ is the quantity and ‘λ’ is the positive integer called ‘decay constant’.

The solution to this problem can be given as

q(t) = q0 e^(-λt)

Here ‘q’ is quantity and the ‘q0’ is the initial value.

Now we will look at some of the measuring rates of the exponential decay

1.     Mean life time: it is the average amount of time, if the element of the decaying quantity
q(t) remains in the set.

It can be represented as decay rate ‘λ’

t = 1/λ

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