지수함수와 로그함수 전체 문제

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 $900?$

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 carrying capacity for the fish population? Justify your answer using the graph of $P.$

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.

Recall the formula for calculating the magnitude of an earthquake, $M=\frac{2}{3}\log \left(\frac{S}{{S}_{0}}\right).$ Show each step for solving this equation algebraically for the seismic moment $S.$

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?

Prove that ${b}^{x}={e}^{x\ln (b)}$ for positive $b\ne 1.$

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.

To the nearest day, how long will it take for half of the Iodine-125 to decay?

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.

Write an exponential model representing the amount of Iodine-125 remaining in the tumor after $t$ days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

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 Radium-226 is $1590$ years. What is the annual 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.

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 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 $5730$ years.)

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?