1. Home
  2. Algebra and trigonometry
  3. Exponential and logarithmic
  4. Exponential and logarithmic

6.7 Exponential and logarithmic models  (Page 4/16)

<< Chapter < Page
  Algebra and trigonometry     Page 4 / 16
Chapter >> Page >

Given the basic exponential growth    equation   A = A 0 e k t , doubling time can be found by solving for when the original quantity has doubled, that is, by solving   2 A 0 = A 0 e k t .

The formula is derived as follows:

2 A 0 = A 0 e k t               2 = e k t Divide by  A 0 .     ln 2 = k t Take the natural logarithm .                 t = ln 2 k Divide by the coefficient of  t .

Thus the doubling time is

t = ln 2 k

Finding a function that describes exponential growth

According to Moore’s Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior.

The formula is derived as follows:

  t = ln 2 k The doubling time formula . 2 = ln 2 k Use a doubling time of two years . k = ln 2 2 Multiply by  k  and divide by 2 . A   = A 0 e ln 2 2 t Substitute  k  into the continuous growth formula .

The function is   A = A 0 e ln 2 2 t .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.

f ( t ) = A 0 e ln 2 3 t

Got questions? Get instant answers now!

Using newton’s law of cooling

Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift    of the generic exponential decay function. This translation leads to Newton’s Law of Cooling    , the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

T ( t ) = a e k t + T s

This formula is derived as follows:

T ( t ) = A b c t + T s T ( t ) = A e ln ( b c t ) + T s Laws of logarithms . T ( t ) = A e c t ln b + T s Laws of logarithms . T ( t ) = A e k t + T s Rename the constant  c   ln   b ,  calling it  k .

Newton’s law of cooling

The temperature of an object,   T , in surrounding air with temperature   T s   will behave according to the formula

T ( t ) = A e k t + T s
where
  • t   is time
  • A   is the difference between the initial temperature of the object and the surroundings
  • k   is a constant, the continuous rate of cooling of the object

Given a set of conditions, apply Newton’s Law of Cooling.

  1. Set   T s   equal to the y -coordinate of the horizontal asymptote (usually the ambient temperature).
  2. Substitute the given values into the continuous growth formula   T ( t ) = A e k t + T s   to find the parameters   A   and   k .
  3. Substitute in the desired time to find the temperature or the desired temperature to find the time.

Using newton’s law of cooling

A cheesecake is taken out of the oven with an ideal internal temperature of   165°F,   and is placed into a   35°F   refrigerator. After 10 minutes, the cheesecake has cooled to   150°F .   If we must wait until the cheesecake has cooled to   70°F   before we eat it, how long will we have to wait?

Because the surrounding air temperature in the refrigerator is 35 degrees, the cheesecake’s temperature will decay exponentially toward 35, following the equation

T ( t ) = A e k t + 35

We know the initial temperature was 165, so   T ( 0 ) = 1 6 5 .

165 = A e k 0 + 35 Substitute  ( 0 , 165 ) .           A = 130 Solve for  A .

We were given another data point,   T ( 1 0 ) = 1 5 0 , which we can use to solve for   k .

                150 = 130 e k 10 + 35 Substitute (10, 150) .                 115 = 130 e k 10 Subtract 35 .                  115 130 = e 10 k Divide by 130 .           ln ( 115 130 ) = 10 k Take the natural log of both sides .                      k = ln ( 115 130 ) 10 = − 0.0123 Divide by the coefficient of  k .

This gives us the equation for the cooling of the cheesecake:   T ( t ) = 1 3 0 e – 0 . 0 1 2 3 t + 3 5 .

Now we can solve for the time it will take for the temperature to cool to 70 degrees.

                            70 = 130 e − 0.0123 t + 35 Substitute in 70 for  T ( t ) .                             35 = 130 e − 0.0123 t Subtract 35 .                     35 130 = e − 0.0123 t Divide by 130 .     ln ( 35 130 ) = − 0.0123 t Take the natural log of both sides                                       t = ln ( 35 130 ) − 0.0123 ≈ 106.68 Divide by the coefficient of  t .

It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to   70°F .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is biology
Hajah Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
what is biology
Victoria Reply
HOW CAN MAN ORGAN FUNCTION
Alfred Reply
the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 6

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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?

Ask