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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 $\text{\hspace{0.17em}}P.$
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 $\text{\hspace{0.17em}}P(t)={P}_{0}{e}^{rt}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}S.$
$\begin{array}{l}\text{}M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right)\hfill \\ \mathrm{log}\left(\frac{S}{{S}_{0}}\right)=\frac{3}{2}M\hfill \\ \text{}\frac{S}{{S}_{0}}={10}^{\frac{3M}{2}}\hfill \\ \text{}S={S}_{0}{10}^{\frac{3M}{2}}\hfill \end{array}$
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,
$\begin{array}{l}\mathrm{ln}(y)=\mathrm{ln}({b}^{x})\hfill \\ \mathrm{ln}(y)=x\mathrm{ln}(b)\hfill \\ {e}^{\mathrm{ln}(y)}={e}^{x\mathrm{ln}(b)}\hfill \\ y={e}^{x\mathrm{ln}(b)}\hfill \end{array}$
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 $\text{\hspace{0.17em}}\mathrm{1.15\%}\text{\hspace{0.17em}}$ 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?
$f(t)=250{e}^{(-0.00914t)};\text{\hspace{0.17em}}$ 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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