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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 .

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

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
  • 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 .

Questions & Answers

Was Psyche rlly a Mortal woman
Elijah Reply
how much for book
Ashley Reply
Whose book?
if it is for this book, psychological trauma and illuminations are each taxed differently depending on your mental preparedness to engage in the subject.
does this apply to the career in applied behavior analysis?
Nicole Reply
Actually I really don't know much about psychology because my age is just 13 ..but I am interested in this subject.I feel it important..to know human behavior and about the world not materially but emotionally.. Can you help me from now to take the courses of psychology from now?
I can try. im only in my 2nd year
read, take notes, study ask questions, study some more. you"ll do just fine. u made it this far.
yes it does.
age isn't a barrier.
same here I am even 12 And I am interested in psychology reading and knowing is ok but what about exams
I have downloaded an app where I used to give exam per day
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Hi can you help me? I need ideas for my thesis title related to psychology
what's the subject?
@nafisa name that app....
psychology quiz
Don't learn psychology, learn what is the possibilities of mind then you will get know what is the psychology in real sens.
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hello guys..go through q youtube and watch some psychological facts about human ....it is really very interesting....
Watch the illusions and be aware about it🤗
all information teaches something, its just not always truth.
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no sir..
I was recently diagnosed schizophrenic. after a lifetime of major depression. time I did hear voices not for long. how does one know if they really schizophrenic?
Cathy Reply
because psychology is the dissection and study of the mind, some find it hard to believe a diagnosis under mental health concerns. if you feel you have been misdiagnosed, go to another doctor. be 100% honest in all questions asked. to lie to your doctor is much like lying to yourself/hurting self
on the contrary, a diagnosed schizophrenic doesn't always have episodes. it may start off very mild and progress over time. it may also be triggered by events/situations/certain states of mind the patient may experience
diagnosed schizophrenics are normally prescribed a medication that can help. if you feel your medication is not working, or that it may make matters worse, discuss this with your doctor so he/she can make adjustments. what works for one person may not work for them all, and so doctors need to know
i want to study about psychology b'coz my father has a mental illness. iwant to help my father to healed from her sicknessed.
please whats the answer
Fortune Reply
answer to what?
what is mean by what?
to the questions ooo i can't find the answers
Can you repeat the question or comment the page you saw it?
why they named it psychology
zayn Reply
telll me
the word "psychology" comes from the Greek words "psyche" meaning life and "logos" meaning "explanation".
Psyche, in Greek mythology was a goddess who comes to represent the human soul's triumph over the misfortunes of life. The term psychology was coined. Psyche, meaning soul or mind and the root "ology" meaning the study of. I hope this helped.
well done
what is chunking
Ritika Reply
Chunking is a process by which individual pieces of an information set are broken down and then grouped together in a meaningful whole.
chunking is to information as syllabus is to word
what dose it means, operant and classic conditioning in short words?
Mikiyas Reply
scientific method, critical thinking.
jeddy Reply
the dealings with the thought process and human behavior.
for acquiring knowledge by experimentation.
study scientific study of mind and behavior.
scientific study of the human mind and behavior.
hey jeddy I have question s for you
hi what is behavioralism ?
what is psychology..?
the science of mental life.
it focuses on understanding and modifying human behaviour
because emotion can't be measured.
marikrizz Reply
only by the person having the emotion.
it is subjective feeling
only placed within levels of itself
and accepted as extended or beginning versions of emotions that are completely free of restrictions
anyone know, (2+2)/2
Emotions can't be measured, but they can be recognized. To be able to recognize certain characteristics that'll give me an idea about an individual's state of mind.
any one knows (2+2)/2
it is 6
anyone (2+2)/2
then more then just recognize but to comprehend the emotions being observed separating your own emotions from the situation currently in.. and rationally gaging the environment.. while then pulling each possible emotional vibe or energy exerted from the human being observed
How does learning influence behaviour
Tumpa Reply
our choice of study molds us and encourages us to practice what we learn. social structure also changes behavior based on the environment and peers. how we learn can be experimental as not everyone lesrns the same. some are more of a hands-on absorber, whereas others may learn better from listening
A confidence from personal knowledge. Not from having been taugbt. knowledge that you learned what was taught.
life is choice, be ready for the best and be prepare for what yet to come in your way(Destiny)
what is development
Momna Reply
In what Context are you referring to
the progression/degression/growth of something. it is interpreted most commonly in psychology as how something is created or formed. ex: When did you develop an interest in psychology?
when refering to physical, cognitive, and social development, these types of development are determined by many circumstances. such as demographics, biology, and personality/character
it is the apprehensions fluency based by the process of configurations to a design based on product/witness series noticeboards to note the stats on order to retail agreements on stated series order.
I'm on my first year of a psychology course, is there anything I should know? anything you wished you knew when you just started psychology ? because I feel like I'm walking into all of this blindly
Lia Reply
I feel exactly the same. Just about to start Psychology and Criminology. feeling slightly nervous 🙈
and everything is online
only for a few weeks I believe. Unless I have read it wrong. I believe that they will try and filter everyone in.
it's different for every country
I start 9/14. Online as well.
I know very little about Psychology
which area are you specialising in
yeah, everthing your just know, that psychology is everthing come out with, mind, behavior and emotion. and believe, once you study it, you will understand, wjy people act like that.
Don't be in the class with pre notions.. yh
after a successful job interview,Mr.Abhishek felt relaxed and calm,he stopped sweating and felt hungry,which part of his nervous system was activated
Babaljeet Reply
what is the Borysenko model and why is it important?
It explains the link between stress and diseases ie.. How long term stress on the mind or body can lead to chronic migraines, ulcers, depression...
I understand there is a link between stress and illness and disease. that when you are in high-stress situations for prolonged periods of time that the brain releases an abundance of stress hormones such as epinephrine, norepinephrine, cortisol, and aldosterone. idk anymore than that.
what is the strees anger
what is difference between illness and disease
anyone knows (2+2)/2
Hy everybody any one who can help me to understand the concept of Structuralism in easy words?
sabiha Reply
If you really considered looking into yourself find the reason of the possibilities or the environment and atmosphere in which you're placed with or without choice in voluntary or involuntary from birth to now, and also consider all existential extensions of yourself including spiritual recognition.
? and compare them to social,judicial, learned events,experiences, influences and dive into there branches? somehow reflects our mechanic motivations and has an effect on who we are and what we represent. ?😲
critical evaluation on attitude change and formation those that even exist
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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