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In this section, you will:
  • Model exponential growth and decay.
  • Use Newton’s Law of Cooling.
  • Use logistic-growth models.
  • Choose an appropriate model for data.
  • Express an exponential model in base e .
Inside a nuclear research reactor.
A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campus (credit: Georgia Tech Research Institute)

We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

Modeling exponential growth and decay

In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:

y = A 0 e k t

where A 0 is equal to the value at time zero, e is Euler’s constant, and k is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth    function in applications involving doubling time , the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.

On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form y = A 0 e k t where A 0 is the starting value, and e is Euler’s constant. Now k is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life    , or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.

In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in [link] and [link] . It is important to remember that, although parts of each of the two graphs seem to lie on the x -axis, they are really a tiny distance above the x -axis.

Graph of y=2e^(3x) with the labeled points (-1/3, 2/e), (0, 2), and (1/3, 2e) and with the asymptote at y=0.
A graph showing exponential growth. The equation is y = 2 e 3 x .
Graph of y=3e^(-2x) with the labeled points (-1/2, 3e), (0, 3), and (1/2, 3/e) and with the asymptote at y=0.
A graph showing exponential decay. The equation is y = 3 e 2 x .

Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude    is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri , measured in kilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is 4.01134972 × 10 13 . So, we could describe this number as having order of magnitude 10 13 .

Questions & Answers

what is the period of cos?
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
A = P(1 + r/n) ^rt
how to solve an expression when equal to zero
Mintah Reply
its a very simple
gave your expression then i solve
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Tomorrow its an revision on factorising and Simplifying...
ok sent the quiz
What is the value of log-1
the value of log1=0
What is the value of i^i
log -1 is 1.36
no I m right
No sister.
no I m right
Joju Reply
jaldi batao
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
Evalute exponential functions
Sujata Reply
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
answer is 2, 3, 4
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
How can I solve for a domain and a codomains in a given function?
Oliver Reply
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proof for set theory
Kwesi Reply
don't you know?
Practice Key Terms 6

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