Question 1

Which of the following equations best models exponential growth, where $$A is $$the amount at time $$t$$, $$A_0 is $$the initial amount, and $$k is $$the growth rate?

Accepted Answer

$$A = A_0$ e$$^{kt}$$

Question 2

A radioactive substance has a half-life of 20 years. If you start with 100 grams, which equation models the amount $$A$$ left after $$t$$ years?

Accepted Answer

$$A = 100 e$$^{-\frac{\ln 2}$${20} t}$$

Question 3

Suppose the population of a city grows according to the model $$P = 5000 e$$^{0.03t}$$, where t$ is in years. What is the population after 10 years?

Accepted Answer

$$P = 5000 e$$^{0.3}$$ \approx 6749$$

Question 4

Which of the following best describes the logistic growth model?

Accepted Answer

$$A = \frac{c}{1 + a e$$^{-bt}$$}$$

Question 5

A psychologist uses the model $$P(t) = 0.4 + \frac{0.4}{1 + 2e$$^{-0.3t}$$} to $$describe the proportion of correct responses after $$t$$ learning trials. What is the limiting value of $$P(t) as$$ $$t$$ approaches infinity?

Accepted Answer

$$0.8$$

Question 6

Newton’s Law of Cooling is given by $$T = C + (T_0$ - C) e$$^{kt}$$. If T_0$ = 100$$, $$C = 20$$, and $$k = -0.1$$, what is the temperature after 10 minutes?

Accepted Answer

$$T = 20 + 80 e$$^{-1}$$ \approx 49.4$$

Question 7

If a substance decays according to $$A = A_0$ e$$^{-kt}$$ and A_0$ = 60$$ grams, $$A = 10$$ grams after $$t$$ years, and $$k = \frac{\ln 2}{28}$$, how long does it take for the substance to decay to 10 grams?

Accepted Answer

$$t = \frac{\ln(6)}{\ln 2 / 28} \approx 72$$ years

Question 8

Which of the following is the correct way to rewrite $$y = ab^x in $$terms of base $$e$$?

Accepted Answer

$$y = a e$$^{x \ln b}$$

Question 9

A city’s population is modeled by $$P = 807 e$$^{0.026t}$$, where t$ is years after 2000. In what year will the population reach 2000 million?

Accepted Answer

$$t = \frac{\ln(2000/807)}{0.026} \approx 36$$,\ so\ year\ 2036$

Question 10

Given the scatter plot of city population versus average walking speed, which type of function is most appropriate for modeling the data if the plot suggests a curve that increases quickly and then levels off?

Accepted Answer

A logarithmic function

Question 11

A cup of coffee is heated to $$90^\circC $$and placed in a $$20^\circC $$room. After 5 minutes, its temperature is $$70^\circC. $$Using Newton’s Law of Cooling, what is the value of $$k in $$the model $$T = 20 + 70 e$$^{kt}$$?

Accepted Answer

$$k = \frac{1}{5} \ln(\frac{70-20}{90-20}) = \frac{1}{5} \ln(\frac{50}{70}) \approx -0.067$$

Question 12

Which of the following statements about exponential decay is correct?

Accepted Answer

The decay rate $$k is $$positive, and the exponent is negative.

Question 13

If a population grows according to $$P = P_0$ e$$^{kt}$$ and doubles every 15 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{15}$$

Question 14

A scientist observes that a bacteria culture triples every 8 hours. If the initial population is $$P_0$$, which equation models the population after $$t$$ hours?

Accepted Answer

$$P = P_0$ e$$^{(\ln 3 / 8)t}$$

Question 15

Which of the following is NOT a characteristic of a logistic growth model?

Accepted Answer

The population grows without bound as $$t$$ increases.

Exponential and Logarithmic Functions: Growth, Decay, and Modeling

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