<< Chapter < Page Chapter >> Page >
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 subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Give me algebra questions
how to send you
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
is there any case that you can have a polynomials with a degree of four?
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
what is linear equation with one unknown 2x+5=3
Joan Reply
I was wrong. I didn't move all constants to the right of the equation.
Adityasuman x= - 1
yas. x= -4
2x=3-5 x=-2/2=-1
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?