6.7 Exponential and logarithmic models  (Page 9/16)

Graph the function.

What is the initial population of fish?

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

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

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

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

To the nearest hour, what is the half-life of the drug?

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.

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

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

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.

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

The half-life of Radium-226 is $1590$ years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

The half-life of Erbium-165 is $\text{10}\text{.4}$ hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.