If a population of bacteria doubles every hour, write an exponential function to model this growth. How does this model illustrate the proportional rate of change?

2.2 Change in Linear and Exponential Functions - Review

Explain how the rate of change in a linear function differs from the rate of change in an exponential function. Provide an example of each.

Describe a real-world scenario where a linear function would be more appropriate to model the situation compared to an exponential function.

Given the linear function $f (x) = 4 x + 1$ and the exponential function $g (x) = 2 \cdot 3^{x}$ , compare their rates of change at $x = 2$ .

How does the graph of a linear function differ from the graph of an exponential function in terms of their slopes?

If a population of bacteria doubles every hour, write an exponential function to model this growth. How does this model illustrate the proportional rate of change?

A population of bacteria doubles every 3 hours. If the initial population is 500 bacteria, write an exponential model to represent the population size after $t$ hours. How many bacteria will there be after 9 hours?

A car depreciates in value by 15% each year. If the initial value of the car is $20,000, write an exponential decay model to represent the value of the car after $t$ years. What will be the value of the car after 5 years?

A certain radioactive substance decays at a rate of 5% per year. If the initial amount of the substance is 100 grams, write an exponential decay model to represent the amount of the substance remaining after $t$ years. How much of the substance will remain after 10 years?

A town's population is growing linearly. In 2010, the population was 50,000, and by 2020, it had grown to 70,000. Write a linear model to represent the population $P (t)$ as a function of $t$ years since 2010. What will be the population in 2030?

10.

A savings account earns interest at an annual rate of 4%, compounded annually. If the initial deposit is $1,000, write an exponential model to represent the amount of money in the account after $t$ years. How much money will be in the account after 5 years?

11.

Describe a real-world scenario where a linear model would be more appropriate than an exponential model. Explain why the linear model is a better fit in this context.

12.

Explain why an exponential model is more suitable for modeling the growth of a viral video’s view count on social media compared to a linear model.

13.

Consider a scenario where a population of rabbits in a closed environment doubles every month. Explain why an exponential model is appropriate for this situation.

14.

A company’s revenue increases by 5% each year. Explain why an exponential model is more appropriate than a linear model for predicting future revenue.

15.

Discuss why a linear model would be inappropriate for modeling the spread of a contagious disease in its early stages.

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AI InstructorAUG 12, 2024, 7:54:33 AM

The exponential function to model the bacteria population doubling every hour is $P (t) = P_{0} \cdot 2^{t}$ , where $P_{0}$ is the initial population and $t$ is the time in hours. This model illustrates the proportional rate of change because the population increases by a factor of 2 for each hour, meaning the rate of change is proportional to the current population size.

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Which of the following functions best represents the population of bacteria doubling every hour?

Select one of the following options:

A.
$P (t) = P_{0} \cdot 2^{t}$
B.
$P (t) = P_{0} + 2 t$
C.
$P (t) = P_{0} \cdot e^{t}$
D.
$P (t) = P_{0} \cdot t^{2}$

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