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What is the initial population of fish?

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

about 1.4 years

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To the nearest whole number, what will the fish population be after 2 years?

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To the nearest tenth, how long will it take for the population to reach 900 ?

about 7.3 years

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What is the carrying capacity for the fish population? Justify your answer using the graph of P .

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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 half-lives; 8.18 minutes

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The formula for an increasing population is given by P ( t ) = P 0 e r t where P 0 is the initial population and r > 0. Derive a general formula for the time t it takes for the population to increase by a factor of M .

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Recall the formula for calculating the magnitude of an earthquake, M = 2 3 log ( S S 0 ) . Show each step for solving this equation algebraically for the seismic moment S .

           M = 2 3 log ( S S 0 ) log ( S S 0 ) = 3 2 M          S S 0 = 10 3 M 2             S = S 0 10 3 M 2

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What is the y -intercept of the logistic growth model y = c 1 + a e r x ? Show the steps for calculation. What does this point tell us about the population?

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Prove that b x = e x ln ( b ) for positive b 1.

Let y = b x for some non-negative real number b such that b 1. Then,

ln ( y ) = ln ( b x ) ln ( y ) = x ln ( b ) e ln ( y ) = e x ln ( b )             y = e x ln ( b )

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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?

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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 3 hours. Round to the nearest milligram.

A = 125 e ( 0.3567 t ) ; A 43 mg

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Using the model found in the previous exercise, find f ( 10 ) and interpret the result. Round to the nearest hundredth.

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For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.

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

about 60 days

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Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t 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.

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A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 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 ( t ) = 250 e ( 0.00914 t ) ; half-life: about 76 minutes

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The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

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The half-life of Erbium-165 is 10 .4 hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

r 0.0667 , So the hourly decay rate is about 6.67 %

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Questions & Answers

hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
Practice Key Terms 6

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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