지수함수와 로그함수 전체 문제

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

Atmospheric pressure $P$ in pounds per square inch is represented by the formula $P=14.7{e}^{-0.21x},$ where $x$ is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of $8.369$ pounds per square inch? (Hint: there are 5280 feet in a mile)

Newton’s Law of Cooling states that the temperature $T$ of an object at any time t can be described by the equation $T={T}_{s}+({T}_{0}-{T}_{s}){e}^{-kt},$ where ${T}_{s}$ is the temperature of the surrounding environment, ${T}_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation $T(t)=68{e}^{-0.0174t}+72.$ To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $t$ years is modeled by the equation $P(t)=\frac{1000}{1+9{e}^{-0.6t}}.$

What is the initial population of fish?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $t$ years is modeled by the equation $P(t)=\frac{1000}{1+9{e}^{-0.6t}}.$

To the nearest tenth, what is the doubling time for the fish population?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $t$ years is modeled by the equation $P(t)=\frac{1000}{1+9{e}^{-0.6t}}.$

To the nearest whole number, what will the fish population be after $2$ years?

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $t$ years is modeled by the equation $P(t)=\frac{1000}{1+9{e}^{-0.6t}}.$

To the nearest tenth, how long will it take for the population to reach $900?$

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?

The formula for an increasing population is given by $P(t)={P}_{0}{e}^{rt}$ where ${P}_{0}$ is the initial population and $r>0.$ Derive a general formula for the time t it takes for the population to increase by a factor of M.

What is the y-intercept of the logistic growth model $y=\frac{c}{1+a{e}^{-rx}}?$ Show the steps for calculation. What does this point tell us about the population?

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?

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ 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.

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

Using the model found in the previous exercise, find $f(10)$ and interpret the result. Round to the nearest hundredth.

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

A scientist begins with $250$ grams of a radioactive substance. After $250$ minutes, the sample has decayed to $32$ grams. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

The half-life of Erbium-165 is $10.4$ hours. What is the hourly decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.

For the following exercises, use this scenario: A tumor is injected with $0.5$ grams of Iodine-125, which has a decay rate of $1.15\%$ per day.

A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was $1350$ bacteria. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest whole number, what is the population size after $3$ hours?

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.

To the nearest whole number, what was the initial population in the culture?

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 $\text{80° F?}$