6.8 Exponential growth and decay  (Page 5/9)

The doubling time for $y=e^{ct}$ is $\left(\text{ln}\left(2\right)\right)\text{/}\left(\text{ln}\left(c\right)\right).$

If you invest $\text{\$}500,$ an annual rate of interest of $3\text{\%}$ yields more money in the first year than a $2.5\text{\%}$ continuous rate of interest.

If you leave a $100\text{°}\text{C}$ pot of tea at room temperature $(25\text{°}\text{C})$ and an identical pot in the refrigerator $(5\text{°}\text{C}),$ with $k=0.02,$ the tea in the refrigerator reaches a drinkable temperature $(70\text{°}\text{C})$ more than $5$ minutes before the tea at room temperature.

If given a half-life of t years, the constant $k$ for $y=e^{kt}$ is calculated by $k=\text{ln}\left(1\text{/}2\right)\text{/}t.$

If a culture of bacteria doubles in $3$ hours, how many hours does it take to multiply by $10?$

If bacteria increase by a factor of $10$ in $10$ hours, how many hours does it take to increase by $100?$

How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is $5730$ years.

If a relic contains $90\text{\%}$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $2000$ years ago)? Note that the half-life of radiocarbon is $5730$ years.

The population of Cairo grew from $5$ million to $10$ million in $20$ years. Use an exponential model to find when the population was $8$ million.

The populations of New York and Los Angeles are growing at $1\text{\%}$ and $1.4\text{\%}$ a year, respectively. Starting from $8$ million (New York) and $6$ million (Los Angeles), when are the populations equal?

Suppose the value of $\text{\$}1$ in Japanese yen decreases at $2\text{\%}$ per year. Starting from $\text{\$}1=\text{¥}250,$ when will $\text{\$}1=\text{¥}1?$

The effect of advertising decays exponentially. If $40\text{\%}$ of the population remembers a new product after $3$ days, how long will $20\text{\%}$ remember it?

If $y=1000$ at $t=3$ and $y=3000$ at $t=4,$ what was $y_0$ at $t=0?$

If $y=100$ at $t=4$ and $y=10$ at $t=8,$ when does $y=1?$

If a bank offers annual interest of $7.5\text{\%}$ or continuous interest of $7.25\text{\%},$ which has a better annual yield?

What continuous interest rate has the same yield as an annual rate of $9\text{\%}?$

If you deposit $\text{\$}5000$ at $8\text{\%}$ annual interest, how many years can you withdraw $\text{\$}500$ (starting after the first year) without running out of money?

You are trying to save $\text{\$}\mathrm{50,000}$ in $20$ years for college tuition for your child. If interest is a continuous $10\text{\%},$ how much do you need to invest initially?

You are cooling a turkey that was taken out of the oven with an internal temperature of $165\text{°}\text{F}.$ After $10$ minutes of resting the turkey in a $70\text{°}\text{F}$ apartment, the temperature has reached $155\text{°}\text{F}\text{.}$ What is the temperature of the turkey $20$ minutes after taking it out of the oven?

You are trying to thaw some vegetables that are at a temperature of $1\text{°}\text{F}\text{.}$ To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $44\text{°}\text{F}.$ You check on your vegetables $2$ hours after putting them in the refrigerator to find that they are now $12\text{°}\text{F}\text{.}$ Plot the resulting temperature curve and use it to determine when the vegetables reach $33\text{°}\text{F}\text{.}$

You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era $(146$ million years to $65$ million years ago), and you find by radiocarbon dating that there is $0.000001\text{\%}$ the amount of radiocarbon. Is this bone from the Cretaceous?

The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $\mathrm{24,000}$ years. If $1$ barrel containing $10\text{kg}$ of plutonium-239 is sealed, how many years must pass until only $10g$ of plutonium-239 is left?

[T] The best-fit exponential curve to the data of the form $P\left(t\right)=a{e}^{bt}$ is given by $P\left(t\right)=2686e^{0.01604t}.$ Use a graphing calculator to graph the data and the exponential curve together.

[T] Find and graph the derivative $y^{\prime }$ of your equation. Where is it increasing and what is the meaning of this increase?

[T] Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?

[T] Find the predicted date when the population reaches $10$ billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

[T] The best-fit exponential curve to the data of the form $P\left(t\right)=a{e}^{bt}$ is given by $P\left(t\right)=35.26e^{0.06407t}.$ Use a graphing calculator to graph the data and the exponential curve together.

[T] Find and graph the derivative $y^{\prime }$ of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?

[T] Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?