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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

if sin15°=√p, express the following in terms of p
Norman Reply
prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch 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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