# 4.6 Exponential and logarithmic models  (Page 9/16)

 Page 9 / 16

Graph the function.

What is the initial population of fish?

To the nearest tenth, what is the doubling time for the fish population?

about $\text{\hspace{0.17em}}1.4\text{\hspace{0.17em}}$ years

To the nearest whole number, what will the fish population be after $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ years?

To the nearest tenth, how long will it take for the population to reach $\text{\hspace{0.17em}}900?$

about $\text{\hspace{0.17em}}7.3\text{\hspace{0.17em}}$ years

What is the carrying capacity for the fish population? Justify your answer using the graph of $\text{\hspace{0.17em}}P.$

## Extensions

A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?

$4\text{\hspace{0.17em}}$ half-lives; $\text{\hspace{0.17em}}8.18\text{\hspace{0.17em}}$ minutes

The formula for an increasing population is given by $\text{\hspace{0.17em}}P\left(t\right)={P}_{0}{e}^{rt}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{P}_{0}\text{\hspace{0.17em}}$ is the initial population and $\text{\hspace{0.17em}}r>0.\text{\hspace{0.17em}}$ Derive a general formula for the time t it takes for the population to increase by a factor of M .

Recall the formula for calculating the magnitude of an earthquake, $\text{\hspace{0.17em}}M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right).$ Show each step for solving this equation algebraically for the seismic moment $\text{\hspace{0.17em}}S.$

What is the y -intercept of the logistic growth model $\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-rx}}?\text{\hspace{0.17em}}$ Show the steps for calculation. What does this point tell us about the population?

Prove that $\text{\hspace{0.17em}}{b}^{x}={e}^{x\mathrm{ln}\left(b\right)}\text{\hspace{0.17em}}$ for positive $\text{\hspace{0.17em}}b\ne 1.$

Let $\text{\hspace{0.17em}}y={b}^{x}\text{\hspace{0.17em}}$ for some non-negative real number $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}b\ne 1.\text{\hspace{0.17em}}$ Then,

## Real-world applications

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

To the nearest hour, what is the half-life of the drug?

Write an exponential model representing the amount of the drug remaining in the patient’s system after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram.

$A=125{e}^{\left(-0.3567t\right)};A\approx 43\text{\hspace{0.17em}}$ mg

Using the model found in the previous exercise, find $\text{\hspace{0.17em}}f\left(10\right)\text{\hspace{0.17em}}$ and interpret the result. Round to the nearest hundredth.

For the following exercises, use this scenario: A tumor is injected with $\text{\hspace{0.17em}}0.5\text{\hspace{0.17em}}$ grams of Iodine-125, which has a decay rate of $\text{\hspace{0.17em}}1.15%\text{\hspace{0.17em}}$ per day.

To the nearest day, how long will it take for half of the Iodine-125 to decay?

about $\text{\hspace{0.17em}}60\text{\hspace{0.17em}}$ days

Write an exponential model representing the amount of Iodine-125 remaining in the tumor after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

A scientist begins with $\text{\hspace{0.17em}}\text{250}\text{\hspace{0.17em}}$ grams of a radioactive substance. After $\text{\hspace{0.17em}}\text{250}\text{\hspace{0.17em}}$ minutes, the sample has decayed to $\text{\hspace{0.17em}}\text{32}\text{\hspace{0.17em}}$ grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

$f\left(t\right)=250{e}^{\left(-0.00914t\right)};\text{\hspace{0.17em}}$ half-life: about $\text{\hspace{0.17em}}\text{76}\text{\hspace{0.17em}}$ minutes

The half-life of Radium-226 is $\text{\hspace{0.17em}}1590\text{\hspace{0.17em}}$ years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

The half-life of Erbium-165 is $\text{\hspace{0.17em}}\text{10}\text{.4}\text{\hspace{0.17em}}$ hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

$r\approx -0.0667,$ So the hourly decay rate is about $\text{\hspace{0.17em}}6.67%$

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
what is variations in raman spectra for nanomaterials
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
Alexandre
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
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
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?
Bob
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?
?
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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