### About Solving Word Problems with Exponential and Logarithmic Functions:

In this section, we will look at how to set up and solve word problems that involve exponential and logarithmic functions. We will see problems that involve exponential growth (doubling time), exponential growth (relative growth rate), radioactive decay, Newton's law of cooling, and logistic growth models.

Test Objectives

- Demonstrate an understanding of exponential and logarithmic functions
- Demonstrate the ability to solve exponential and logarithmic equations
- Demonstrate the ability to solve a word problem that involves an exponential function
- Demonstrate the ability to solve a word problem that involves a logarithmic function

#1:

Instructions: Solve each word problem.

Exponential Growth (Doubling Time)

If the initial size of a population is n_{0} and the doubling time is a, then the size of the population at time t is given by: $$\large{n(t)=n_{0}\cdot 2^{\frac{t}{a}}}$$ where a and t are measured in the same units of time (seconds, minutes, hours, days, years, and so on).

Under ideal conditions, a certain bacteria population doubles every five hours. Initially, there are 10 bacteria in a colony.

a) Find a model for the bacteria population after t hours.

b) How many bacteria are in the colony after 20 hours?

c) After how many hours will the bacteria count reach 1,280?

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#2:

Instructions: Solve each word problem.

Exponential Growth (Relative Growth Rate)

A population that experiences exponential growth increases according to the model: $$\large{n(t)=n_{0}\cdot e^{rt}}$$ n(t) is the population at time t, n_{0} is the initial size of the population, r is the relative rate of growth (expressed as a proportion of the population), and t is the time.

A lab contains a bacteria culture with an initial count of 150. After some time, a scientist obtains a sample count of bacteria in the culture and determines that the relative rate of growth is 18% per hour.

a) Find a function that models the number of bacteria after t hours.

b) What is the estimated count after 5 hours?

c) After how many hours will the bacteria count reach 1,000?

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#3:

Instructions: Solve each word problem.

Radioactive Decay Model

If m_{0} is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is given by: $$m(t)=m_{0}e^{-rt}$$ $$r=\frac{\ln(2)}{h}$$ r is known as the relative decay rate.

Polonium-210 has a half-life of 140 days. A researcher in a lab obtains a sample of Polonium-210 that has a mass of 500 milligrams.

a) Find a function that models the mass remaining after t days.

b) Find the mass remaining after two years (ignore leap years).

c) How long will it take for the sample to decay to a mass of 5 milligrams?

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#4:

Instructions: Solve each word problem.

Newton's Law of Cooling

If D_{0} is the initial temperature difference between an object and its surroundings, and if its surroundings have temperature T_{s}, then the temperature of the object at time t is given by: $$T(t)=T_s + D_0 \cdot e^{-kt}$$ where k is a positive constant that depends on the type of object.

Note: in this formula, the capital letter "T" is used for temperature, while the lowercase letter "t" is used for time.

A cup of coffee has a temperature of 195°F and is placed in a room that has a temperature of 68°F. After 12 minutes, the temperature of the coffee is 142°F.

a) Find a function that models the temperature of the coffee at time t.

b) Find the temperature of the coffee after 18 minutes. Round to the nearest degree.

c) After how long will the coffee have cooled to 95°F? Round to the nearest minute.

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#5:

Instructions: Solve each word problem.

Logistic Growth Models

A model for limited logistic growth is given by: $$f(t)=\frac{c}{1 + ae^{-bt}}$$ Where a, b, and c are constants, with c > 0 and b > 0.

Evergreen High School currently has 5,000 students. A student returned from a weekend trip with a persistent and easily transmissible influenza strain. The spread of the infection is given by the following: $$f(t)=\frac{5{,}000}{1 + 4{,}999e^{-0.8t}}, t ≥ 0$$ Where f(t) is the total number of Evergreen students infected after t days. Evergreen High School has a policy of canceling classes when 57% or more students are infected with an outbreak.

a) How many students are infected after 2, 5, and 10 days?

b) After how many days will Evergreen cancel classes?

c) What is the limiting size of f(t)?

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}n(t)=10 \cdot 2^{\frac{t}{5}}$$ n(t) is the number of bacteria after t hours.

$$b)\hspace{.2em}n(20)=10 \cdot 2^{\frac{20}{5}}=160$$ After 20 hours, the number of bacteria is 160.

$$c)\hspace{.2em}1{,}280=10 \cdot 2^{\frac{t}{5}}$$ $$t=35$$The bacteria count reaches 1,280 after 35 hours.

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#2:

Solutions:

$$a)\hspace{.2em}n(t)=150e^{0.18t}$$ n(t) is the number of bacteria after t hours.

$$b)\hspace{.2em}n(5)=150e^{0.18(5)}≈ 369$$ After 5 hours, the number of bacteria is about 369.

$$c)\hspace{.2em}1{,}000=150e^{0.18t}$$ $$t=\frac{\ln\left(\frac{20}{3}\right)}{0.18}≈ 10.54$$ The bacteria count reaches 1,000 in about 10.54 hours.

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#3:

Solutions:

$$a)\hspace{.2em}m(t)=500e^{\large{-\frac{\ln(2)}{140}t}}$$ m(t) is the mass remaining after t days.

$$b)\hspace{.2em}m(730) ≈ 13.47$$ After 2 years or 730 days, there will be about 13.47 milligrams.

$$c)\hspace{.2em}500e^{\large{-\frac{\ln(2)}{140}t}}=5$$ $$t=\frac{140\ln(100)}{\ln(2)}≈ 930$$ The time required for the sample to decay to 5 milligrams is about 930 days.

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#4:

Solutions:

$$a)\hspace{.2em}T(t)=68 + 127e^{-0.045t}$$ T(t) is our model that represents the temperature after t minutes.

If you want an exact value for the -k part: $$T(t)=68 + 127e^{\frac{\ln\left(\frac{74}{127}\right)}{12}t}$$ $$\frac{\ln\left(\frac{74}{127}\right)}{12}≈ -0.045$$

$$b)\hspace{.2em}T(18)=68 + 127e^{-0.045(18)}≈ 124$$ After 18 minutes, the coffee has a temperature of about 124°F.

$$c)\hspace{.2em}95=68 + 127e^{-0.045t}$$ After about 34 minutes, the coffee has a temperature of 95°F.

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#5:

Solutions:

a) After 2 days, about 5 students are infected, after 5 days, about 54 students are infected, and after 10 days, about 1,868 students are infected.

b) After about 11 days, at least 57% of the students are infected, and Evergreen will cancel classes.

c) The total number of students at Evergreen High School, which is 5,000 is the limiting size of f(t).