# 4.6 Exponential and logarithmic models  (Page 4/16)

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Given the basic exponential growth    equation $\text{\hspace{0.17em}}A={A}_{0}{e}^{kt},$ doubling time can be found by solving for when the original quantity has doubled, that is, by solving $\text{\hspace{0.17em}}2{A}_{0}={A}_{0}{e}^{kt}.$

The formula is derived as follows:

Thus the doubling time is

$t=\frac{\mathrm{ln}2}{k}$

## 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:

The function is $\text{\hspace{0.17em}}A={A}_{0}{e}^{\frac{\mathrm{ln}2}{2}t}.$

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.

$f\left(t\right)={A}_{0}{e}^{\frac{\mathrm{ln}2}{3}t}$

## 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

$T\left(t\right)=a{e}^{kt}+{T}_{s}$

This formula is derived as follows:

## Newton’s law of cooling

The temperature of an object, $\text{\hspace{0.17em}}T,$ in surrounding air with temperature $\text{\hspace{0.17em}}{T}_{s}\text{\hspace{0.17em}}$ will behave according to the formula

$T\left(t\right)=A{e}^{kt}+{T}_{s}$
where
• $t\text{\hspace{0.17em}}$ is time
• $A\text{\hspace{0.17em}}$ is the difference between the initial temperature of the object and the surroundings
• $k\text{\hspace{0.17em}}$ is a constant, the continuous rate of cooling of the object

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

1. Set $\text{\hspace{0.17em}}{T}_{s}\text{\hspace{0.17em}}$ equal to the y -coordinate of the horizontal asymptote (usually the ambient temperature).
2. Substitute the given values into the continuous growth formula $\text{\hspace{0.17em}}T\left(t\right)=A{e}^{k}{}^{t}+{T}_{s}\text{\hspace{0.17em}}$ to find the parameters $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}k.$
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 $\text{\hspace{0.17em}}\text{165°F,}\text{\hspace{0.17em}}$ and is placed into a $\text{\hspace{0.17em}}35°F\text{\hspace{0.17em}}$ refrigerator. After 10 minutes, the cheesecake has cooled to $\text{\hspace{0.17em}}\text{150°F}\text{.}\text{\hspace{0.17em}}$ If we must wait until the cheesecake has cooled to $\text{\hspace{0.17em}}\text{70°F}\text{\hspace{0.17em}}$ 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

$T\left(t\right)=A{e}^{kt}+35$

We know the initial temperature was 165, so $\text{\hspace{0.17em}}T\left(0\right)=165.$

We were given another data point, $\text{\hspace{0.17em}}T\left(10\right)=150,$ which we can use to solve for $\text{\hspace{0.17em}}k.$

This gives us the equation for the cooling of the cheesecake: $\text{\hspace{0.17em}}T\left(t\right)=130{e}^{–0.0123t}+35.$

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

It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to $\text{\hspace{0.17em}}\text{70°F}\text{.}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
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what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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How we are making nano material?
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LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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