지수함수와 로그함수 전체 문제

For the following exercises, use this scenario: A biologist recorded a count of $360$ bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.

Rounding to six decimal places, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of $100^\circ$ Fahrenheit was taken off the stove to cool in a $\text{69° F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{95° F}.$

Use Newton’s Law of Cooling to write a formula that models this situation.

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of $100^\circ$ Fahrenheit was taken off the stove to cool in a $\text{69° F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{95° F}.$

To the nearest minute, how long will it take the soup to cool to $\text{80° F?}$

For the following exercises, use this scenario: A pot of warm soup with an internal temperature of $100^\circ$ Fahrenheit was taken off the stove to cool in a $\text{69° F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{95° F}.$

To the nearest degree, what will the temperature be after $2$ and a half hours?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

Write a formula that models this situation.

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

To the nearest degree, what will the temperature be after 50 minutes?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{165°F}$ and is allowed to cool in a $\text{75°F}$ room. After half an hour, the internal temperature of the turkey is $\text{145°F}.$

To the nearest minute, how long will it take the turkey to cool to $\text{110° F?}$

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: ${10}^{-10} \frac{W}{{m}^{2}},$ Vacuum: ${10}^{-4}\frac{W}{{m}^{2}},$ Jet: ${10}^{2} \frac{W}{{m}^{2}}$

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

Recall the formula for calculating the magnitude of an earthquake, $M=\frac{2}{3}\log \left(\frac{S}{{S}_{0}}\right).$ One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

How many people started the rumor?

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

To the nearest whole number, how many people will have heard the rumor after 3 days?

For the following exercises, use this scenario: The equation $N(t)=\frac{500}{1+49{e}^{-0.7t}}$ models the number of people in a town who have heard a rumor after t days.

As $t$ increases without bound, what value does $N(t)$ approach? Interpret your answer.

For the following exercise, choose the correct answer choice.

A doctor injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

• ⓐ $f(t)=13{(0.0805)}^{t}$
• ⓑ $f(t)=13{e}^{0.9195t}$
• ⓒ $f(t)=13{e}^{(-0.0839t)}$
• ⓓ $f(t)=\frac{4.75}{1+13{e}^{-0.83925t}}$

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

What might a scatterplot of data points look like if it were best described by a logarithmic model?

What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?