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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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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