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The population of a city is modeled by the equation P ( t ) = 256 , 114 e 0.25 t where t 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 5.45 years

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Find the inverse function f 1 for the exponential function f ( x ) = 2 e x + 1 5.

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Find the inverse function f 1 for the logarithmic function f ( x ) = 0.25 log 2 ( x 3 + 1 ) .

f 1 ( x ) = 2 4 x 1 3

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Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17 % each hour.

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

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Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 24 hours. Round to the nearest hundredth of a gram.

f ( t ) = 300 ( 0.83 ) t ; f ( 24 ) 3.43 g

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

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

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How many minutes will it take the soup to cool to 85°F?

about 45 minutes

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For the following exercises, use this scenario: The equation N ( t ) = 1200 1 + 199 e 0.625 t models the number of people in a school who have heard a rumor after t days.

How many people started the rumor?

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

about 8.5 days

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What is the carrying capacity?

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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

Graph of the table’s values.
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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
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Find a formula for an exponential equation that goes through the points ( 2 , 100 ) and ( 0 , 4 ) . Then express the formula as an equivalent equation with base e.

y = 4 ( 0.2 ) x ; y = 4 e -1 .609438 x

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Fitting Exponential Models to Data

What is the carrying capacity for a population modeled by the logistic equation P ( t ) = 250 , 000 1 + 499 e 0.45 t ? What is the initial population for the model?

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The population of a culture of bacteria is modeled by the logistic equation P ( t ) = 14 , 250 1 + 29 e 0.62 t , where t is in days. To the nearest tenth, how many days will it take the culture to reach 75 % of its carrying capacity?

about 7.2 days

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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.

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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