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A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is $\text{\hspace{0.17em}}\text{573}0\text{\hspace{0.17em}}$ years.)
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was $\text{\hspace{0.17em}}1350\text{\hspace{0.17em}}$ bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after $\text{\hspace{0.17em}3\hspace{0.17em}}$ hours?
$f(t)=1350{e}^{(0.03466t)};\text{\hspace{0.17em}}$ after 3 hours: $\text{\hspace{0.17em}}P(180)\approx 691,200$
For the following exercises, use this scenario: A biologist recorded a count of $\text{\hspace{0.17em}}360\text{\hspace{0.17em}}$ bacteria present in a culture after $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ minutes and $\text{\hspace{0.17em}}1000\text{\hspace{0.17em}}$ bacteria present after $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ minutes.
To the nearest whole number, what was the initial population in the culture?
Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?
$f(t)=256{e}^{(0.068110t)};\text{\hspace{0.17em}}$ doubling time: about $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ minutes
For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of $\text{\hspace{0.17em}}\text{100\xb0}\text{\hspace{0.17em}}$ Fahrenheit was taken off the stove to cool in a $\text{\hspace{0.17em}}\text{69\xb0F}\text{\hspace{0.17em}}$ room. After fifteen minutes, the internal temperature of the soup was $\text{\hspace{0.17em}}\text{95\xb0F}\text{.}$
Use Newton’s Law of Cooling to write a formula that models this situation.
To the nearest minute, how long will it take the soup to cool to $\text{\hspace{0.17em}}\text{80\xb0F?}$
about $\text{\hspace{0.17em}}\text{88}\text{\hspace{0.17em}}$ minutes
To the nearest degree, what will the temperature be after $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ and a half hours?
For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{\hspace{0.17em}}\text{165\xb0F}\text{\hspace{0.17em}}$ and is allowed to cool in a $\text{\hspace{0.17em}}\text{75\xb0F}\text{\hspace{0.17em}}$ room. After half an hour, the internal temperature of the turkey is $\text{\hspace{0.17em}}\text{145\xb0F}\text{.}$
Write a formula that models this situation.
$T(t)=90{e}^{(-0.008377t)}+75,$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in minutes.
To the nearest degree, what will the temperature be after 50 minutes?
To the nearest minute, how long will it take the turkey to cool to $\text{\hspace{0.17em}}\text{110\xb0F?}$
about $\text{\hspace{0.17em}}\text{113}\text{\hspace{0.17em}}$ minutes
For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.
$\mathrm{log}\left(x\right)=1.5;\text{\hspace{0.17em}}x\approx 31.623$
Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: $\text{\hspace{0.17em}}{10}^{-10}\frac{W}{{m}^{2}},$ Vacuum: $\text{\hspace{0.17em}}{10}^{-4}\frac{W}{{m}^{2}},$ Jet: $\text{\hspace{0.17em}}{10}^{2}\frac{W}{{m}^{2}}$
Recall the formula for calculating the magnitude of an earthquake, $\text{\hspace{0.17em}}M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right).\text{\hspace{0.17em}}$ One earthquake has magnitude $\text{\hspace{0.17em}}\text{3}.\text{9}\text{\hspace{0.17em}}$ on the MMS scale. If a second earthquake has $\text{\hspace{0.17em}}\text{75}0\text{\hspace{0.17em}}$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.
MMS magnitude: $\text{\hspace{0.17em}}5.82$
For the following exercises, use this scenario: The equation $\text{\hspace{0.17em}}N\left(t\right)=\frac{500}{1+49{e}^{-0.7t}}\text{\hspace{0.17em}}$ models the number of people in a town who have heard a rumor after t days.
How many people started the rumor?
To the nearest whole number, how many people will have heard the rumor after 3 days?
$N(3)\approx 71$
As $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ increases without bound, what value does $\text{\hspace{0.17em}}N\left(t\right)\text{\hspace{0.17em}}$ approach? Interpret your answer.
For the following exercise, choose the correct answer choice.
A doctor and injects a patient with $\text{\hspace{0.17em}}13\text{\hspace{0.17em}}$ milligrams of radioactive dye that decays exponentially. After $\text{\hspace{0.17em}}12\text{\hspace{0.17em}}$ minutes, there are $\text{\hspace{0.17em}}4.75\text{\hspace{0.17em}}$ milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?
C
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