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

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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 $\text{\hspace{0.17em}}6.67%$

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