# 4.8 Fitting exponential models to data  (Page 11/12)

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The population of a city is modeled by the equation $\text{\hspace{0.17em}}P\left(t\right)=256,114{e}^{0.25t}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

about $\text{\hspace{0.17em}}5.45\text{\hspace{0.17em}}$ years

Find the inverse function $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ for the exponential function $\text{\hspace{0.17em}}f\left(x\right)=2\cdot {e}^{x+1}-5.$

Find the inverse function $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ for the logarithmic function $\text{\hspace{0.17em}}f\left(x\right)=0.25\cdot {\mathrm{log}}_{2}\left({x}^{3}+1\right).$

${f}^{-1}\left(x\right)=\sqrt[3]{{2}^{4x}-1}$

## Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes $\text{\hspace{0.17em}}300\text{\hspace{0.17em}}$ milligrams of a therapeutic drug that decays by about $\text{\hspace{0.17em}}17%\text{\hspace{0.17em}}$ each hour.

To the nearest minute, 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 $\text{\hspace{0.17em}}24\text{\hspace{0.17em}}$ hours. Round to the nearest hundredth of a gram.

$f\left(t\right)=300{\left(0.83\right)}^{t};f\left(24\right)\approx 3.43\text{ }\text{ }g$

For the following exercises, use this scenario: A soup with an internal temperature of $\text{\hspace{0.17em}}\text{350°}\text{\hspace{0.17em}}$ Fahrenheit was taken off the stove to cool in a $\text{\hspace{0.17em}}\text{71°F}\text{\hspace{0.17em}}$ room. After fifteen minutes, the internal temperature of the soup was $\text{\hspace{0.17em}}\text{175°F}\text{.}$

Use Newton’s Law of Cooling to write a formula that models this situation.

How many minutes will it take the soup to cool to $\text{\hspace{0.17em}}\text{85°F?}$

about $\text{\hspace{0.17em}}45\text{\hspace{0.17em}}$ minutes

For the following exercises, use this scenario: The equation $\text{\hspace{0.17em}}N\left(t\right)=\frac{1200}{1+199{e}^{-0.625t}}\text{\hspace{0.17em}}$ models the number of people in a school who have heard a rumor after $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ days.

How many people started the rumor?

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

about $\text{\hspace{0.17em}}8.5\text{\hspace{0.17em}}$ days

What is the carrying capacity?

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

 x f(x) 1 3.05 2 4.42 3 6.4 4 9.28 5 13.46 6 19.52 7 28.3 8 41.04 9 59.5 10 86.28

exponential

 x f(x) 0.5 18.05 1 17 3 15.33 5 14.55 7 14.04 10 13.5 12 13.22 13 13.1 15 12.88 17 12.69 20 12.45

Find a formula for an exponential equation that goes through the points $\text{\hspace{0.17em}}\left(-2,100\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,4\right).\text{\hspace{0.17em}}$ Then express the formula as an equivalent equation with base e.

$y=4{\left(0.2\right)}^{x};\text{\hspace{0.17em}}$ $y=4{e}^{\text{-1}\text{.609438}x}$

## Fitting Exponential Models to Data

What is the carrying capacity for a population modeled by the logistic equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{250,000}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}499{e}^{-0.45t}}?\text{\hspace{0.17em}}$ What is the initial population for the model?

The population of a culture of bacteria is modeled by the logistic equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{14,250}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}29{e}^{-0.62t}},$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in days. To the nearest tenth, how many days will it take the culture to reach $\text{\hspace{0.17em}}75%\text{\hspace{0.17em}}$ of its carrying capacity?

about $\text{\hspace{0.17em}}7.2\text{\hspace{0.17em}}$ days

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this