Question 1

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

Accepted Answer

$$f(t) = A_0$ e$$^{kt}$$

Question 2

A scientist is studying a radioactive substance with a half-life of 20 years. If the initial mass is 80 grams, which exponential decay model describes the amount remaining after $$t$$ years?

Accepted Answer

$$A(t) = 80 e$$^{-0.0347t}$$

Question 3

Given the exponential growth model $$P(t) = 807e$$^{kt}$$ for Africa's population (in millions), and knowing that P(11) = 1052$, what is the value of k$ to three decimal places?

Accepted Answer

$$k = 0.024$$

Question 4

A city’s population grows according to $$P(t) = 1000e$$^{0.02t}$$. What will the population be after 35 years? (Round to the nearest whole number.)

Accepted Answer

$$P(35) = 2019$$

Question 5

The half-life of a substance is 28 years. If you start with 60 grams, how much remains after 56 years?

Accepted Answer

$$15$$ grams

Question 6

Which of the following best describes the logistic growth model?

Accepted Answer

$$f(t) = \frac{c}{1 + ae$$^{-bt}$$}$$

Question 7

In a learning theory project, the proportion of correct responses is modeled by $$f(t) = 0.8 - 0.4e$$^{-0.2t}$$. What is the limiting value of f(t$$)$$ as t$ approaches infinity?

Accepted Answer

$$0.8$$

Question 8

Given the model $$f(t) = 0.8 - 0.4e$$^{-0.2t}$$, what is the proportion of correct responses before any learning trials (t = 0$)?

Accepted Answer

$$0.4$$

Question 9

A city’s population is modeled by $$P(t) = 500e$$^{0.03t}$$. In how many years will the population double?

Accepted Answer

$$t = 23.1$$

Question 10

Which type of function would best model the relationship between city population and average walking speed, based on the scatter plot in Table 4.7?

Accepted Answer

Logarithmic

Question 11

Rewrite $$y = 4(7.8)^x in $$terms of base $$e$$ and round the exponent coefficient to three decimal places.

Accepted Answer

$$y = 4e$$^{2.054x}$$

Question 12

A sample of strontium-90 decays according to $$A(t) = 60e$$^{-kt}$$, where k$ is the decay constant. If the half-life is 28 years, what is the value of k$ (rounded to three decimal places)?

Accepted Answer

$$k = 0.025$$

Question 13

If a population grows according to $$P(t) = P_0e$$^{kt}$$ and k$$ \u003c 0$$, what does this indicate about the population?

Accepted Answer

The population is decaying exponentially.

Question 14

A scientist observes that a bacteria culture triples every 5 hours. Which equation models the population $$P(t)$$ after $$t$$ hours, if the initial population is $$P_0$$?

Accepted Answer

$$P(t) = P_0e$$^{\frac{\ln 3}$${5}t}$$

Question 15

A radioactive isotope decays to 25% of its original amount in 56 years. What is its half-life?

Accepted Answer

$$28$$ years

Exponential and Logarithmic Functions: Growth, Decay, and Modeling

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