6.7 Exponential and Logarithmic Models

A total of $\$ 10,000$ was invested in a college fund for a new grandchild.

ⓐ If the interest rate is $5 \% ,$ how much will be in the account in 18 years by each method of compounding?

ⓑ compound quarterly

ⓒ compound monthly

ⓓ compound continuously

Chris is a researcher at the Center for Disease Control and Prevention and he is trying to understand the behavior of a new and dangerous virus. He starts his experiment with 100 of the virus that grows at a rate of 25% per hour. He will check on the virus in 24 hours. How many viruses will he find?

Use logarithmic models in applications.

Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through earphones with intensity $10^{-2}$ watts per square inch?

In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused $108 million dollars of damage. Compare the intensities of the two earthquakes.

The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, $t .$

The half-life of plutonium-244 is 80,000,000 years. Find a function that gives the amount of plutonium-244 remaining as a function of time, measured in years.

A bone fragment is found that contains 20% of its original carbon-14. To the nearest year, how old is the bone?

Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take more or less than 230 years until only 1 milligram remains?

According to Moore’s Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior.

Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.

A cheesecake is taken out of the oven with an ideal internal temperature of $\text{165}°\text{F},$ and is placed into a $35°F$ refrigerator. After 10 minutes, the cheesecake has cooled to $\text{150}°\text{F} .$ If we must wait until the cheesecake has cooled to $\text{70}°\text{F}$ before we eat it, how long will we have to wait?

A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the temperature has risen to 45 degrees. How long will it take for the temperature to rise to 60 degrees?

An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of communicable diseases. And, clearly, there is a maximum value for the number of people…

Using the model in the example, estimate the number of cases of flu on day 15.

Does a linear, exponential, or logarithmic model best fit the data in the table? Find the model.

Change the function $y = 2.5 \left(3.1\right)^{x}$ so that this same function is written in the form $y = A_{0} e^{k x} .$

Change the function $y = 3 \left(0.5\right)^{x}$ to one having $e$ as the base.

With what kind of exponential model would half-life be associated? What role does half-life play in these models?

With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

Use the logistic growth model $f \left(x\right) = \frac{150}{1 + 8 e^{- 2 x}} .$

Find and interpret $f \left(0\right) .$ Round to the nearest tenth.

Use the logistic growth model $f \left(x\right) = \frac{150}{1 + 8 e^{- 2 x}} .$

Find the carrying capacity.

Use the logistic growth model $f \left(x\right) = \frac{150}{1 + 8 e^{- 2 x}} .$

Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.

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

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

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

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?

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 $S .$

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

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.

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?

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 $\text{250}$ grams of a radioactive substance. After $\text{250}$ minutes, the sample has decayed to $\text{32}$ 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?

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 $\text{10} .\text{4}$ hours. What is the hourly decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.

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 a…

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?

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

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

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

Write a formula that models this situation.

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

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

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 $\text{3} . \text{9}$ on the MMS scale. If a second earthquake has $\text{75} 0$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

Use this scenario: The equation $N \left(t\right) = \frac{500}{1 + 49 e^{- 0.7 t}}$ 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?

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?

1. ⓐ $f \left(t\right) = 13 \left(0.0805\right)^{t}$
2. ⓑ $f \left(t\right) = 13 e^{0.9195 t}$
3. ⓒ $f \left(t\right) = 13 e^{\left(- 0.0839 t\right)}$
4. ⓓ $f \left(t\right) = \frac{4.75}{1 + 13 e^{- 0.83925 t}}$