지수적 증가와 감소

True or False? If true, prove it. If false, find the true answer.

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

If you leave a $100^\circ \text{C}$ pot of tea at room temperature $(25^\circ \text{C})$ and an identical pot in the refrigerator $(5^\circ \text{C}),$ with $k=0.02,$ the tea in the refrigerator reaches a drinkable temperature $(70^\circ \text{C})$ 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 $k=\ln \,(1/2)/t.$

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 $90\%$ 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 $1\%$ and $1.4\%$ a year, respectively. Starting from $8$ million (New York) and $6$ million (Los Angeles), when are the populations equal? Round your answer to a whole number of years.

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

For the following exercises, use $y={y}_{0}{e}^{kt}.$

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

For the following exercises, use $y={y}_{0}{e}^{kt}.$

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

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

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

You are cooling a turkey that was taken out of the oven with an internal temperature of $165^\circ \text{F}.$ After $10$ minutes of resting the turkey in a $70^\circ \text{F}$ apartment, the temperature has reached $155^\circ \text{F}.$ 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 $1^\circ \text{F}.$ To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $44^\circ \text{F}.$ You check on your vegetables $2$ hours after putting them in the refrigerator to find that they are now $12^\circ \text{F}.$ Plot the resulting temperature curve and use it to determine when the vegetables reach $33^\circ \text{F}.$

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 $0.000001\%$ 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 $24,000$ 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 $P(t)=a{e}^{bt}$ is given by $P(t)=2686{e}^{0.01604t}.$ Use a graphing calculator to graph the data and the exponential curve together.

[T] Find and graph the derivative ${y}^{'}$ of your equation. Where is it increasing and what is the meaning of this increase?