1. Home
  2. Precalculus
  3. Exponential and logarithmic
  4. Exponential and logarithmic

4.7 Exponential and logarithmic models  (Page 4/16)

<< Chapter < Page
  Precalculus     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

how do you get the 2/50
Abba Reply
number of sport play by 50 student construct discrete data
Aminu Reply
width of the frangebany leaves on how to write a introduction
Theresa Reply
Solve the mean of variance
Veronica Reply
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.
kenneth
what is error
Yakuba Reply
Is mistake done to something
Vutshila
Hy
anas
hy
What is the life teble
anas
hy
Jibrin
statistics is the analyzing of data
Tajudeen Reply
what is statics?
Zelalem Reply
how do you calculate mean
Gloria Reply
diveving the sum if all values
Shaynaynay
let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?
Fisaye Reply
what is statistics
Akhisani Reply
data collected all over the world
Shaynaynay
construct a less than and more than table
Imad Reply
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Aschalew Reply
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?
Akshay Reply
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits
Akshay
1
Bright
Sorry i want to learn more about this question
Bright
Someone help
Bright
a= 0.20233 b=0.3384
Sufiyan
a
Shaynaynay
How do I interpret level of significance?
Mohd Reply
It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value
Shivam
how skewness and kurtosis are used in statistics
Owen Reply
yes what is it
Taneeya
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, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask