Q: The half-life of a radioactive isotope is 20 years. If you start with 80 grams, how much will remain after 60 years?

$$80 \times \left(\frac{1}{2}\$$right)^3$$ = 10$$ grams

Question 1

A population of rabbits in a park is initially 200 and increases at a constant rate of 8% per year. Which of the following equations models the population $$P$$ after $$t$$ years?

Accepted Answer

$$P = 200(1.08)^t$$

Question 2

A scientist places 500 bacteria in a dish. The bacteria triple every 2 hours. How many bacteria will there be after 6 hours?

Accepted Answer

$$500 	imes 3^3$ = 13,500$$

Question 3

The half-life of a radioactive isotope is 20 years. If you start with 80 grams, how much will remain after 60 years?

Accepted Answer

$$80 	imes \left(\frac{1}{2}\$$right)^3$$ = 10$$ grams

Question 4

A rumor spreads in a school of 1200 students according to the logistic model $$N(t) = \frac{1200}{1 + 23e$$^{-0.5t}$$}. $$How many students have heard the rumor at $$t = 0$$?

Accepted Answer

$$N(0) = \frac{1200}{1 + 23} = 50$$

Question 5

Which of the following best describes the difference between exponential and logistic growth models?

Accepted Answer

Exponential growth is unrestricted, while logistic growth includes a maximum sustainable population.

Question 6

Given the exponential regression equation $$y = 66.99(1.15)^x$$, what is the predicted value of $$y$$ when $$x = 11$$?

Accepted Answer

$$y = 66.99(1.15$$)^{11}$$ \approx 311.66$$

Question 7

A city’s population is modeled by $$P(t) = 100,000(1.03)^t$$, where $$t is $$the number of years since 2020. What is the population in 2025?

Accepted Answer

$$100,000(1.03)^5$ \approx 115,927$$

Question 8

If a population follows the model $$P = P_0$(1 + r)^t$$, what does the parameter $$r$$ represent?

Accepted Answer

The constant percentage rate of change, expressed as a decimal.

Question 9

A radioactive substance decays according to $$M(t) = 6\left(\frac{1}{2}\$$right)^{t/30}$$, where t$ is in days. How long will it take for only 1 gram to remain?

Accepted Answer

$$t = 30 \cdot \log_{1/2}\left(\frac{1}{6}\right) \approx 77.55$$ days

Question 10

Which real-world scenario is best modeled by an exponential decay function?

Accepted Answer

The amount of a radioactive isotope decreasing over time.

Question 11

A population of fish in a lake is modeled by $$P(t) = 500(1.12)^t. If $$the population reaches 1,000, in how many years will this occur? (Round to the nearest year.)

Accepted Answer

$$t = \frac{\log(2)}{\log(1.12)} \approx 6$$ years

Question 12

Which of the following is NOT a characteristic of exponential growth?

Accepted Answer

The function eventually levels off at a maximum value.

Question 13

A logistic growth model is used to describe the spread of a rumor in a school. Which feature distinguishes logistic growth from exponential growth?

Accepted Answer

Logistic growth has a carrying capacity or maximum limit.

Question 14

Suppose a population is modeled by $$P(t) = 1000(0.95)^t. $$What does the factor $$0.95$$ represent?

Accepted Answer

A 5% decrease per time period.

Question 15

A scientist uses exponential regression to fit the model $$y = ab^x to $$data. If $$a = 50$$ and $$b = 1.2$$, what is the predicted value when $$x = 4$$?

Accepted Answer

$$50 	imes (1.2)^4$ = 103.68$$

Question 16

Which of the following situations is best modeled by a logistic function?

Accepted Answer

The spread of a rumor in a closed population.

Question 17

If a population doubles every 5 years, what is the annual percentage growth rate $$r ($$rounded to the nearest percent)?

Accepted Answer

$$r = (2$$)^{1/5}$$ - 1 \approx 0.15 or$$ $$15\%$$

Question 18

A population is modeled by $$P(t) = 5000e$$^{0.04t}$$. What is the continuous growth rate?

Accepted Answer

$$4\%$$

Question 19

Which of the following best explains why exponential models are used for radioactive decay?

Accepted Answer

The rate of decay is proportional to the current amount of substance.

Exponential and Logarithmic Functions: Modeling Growth and Decay

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