Question 1

Which of the following equations best models exponential growth, as depicted in the first image?

Accepted Answer

$$A = A_0$ e$$^{kt}$$ where k$$ \u003e 0$$

Question 2

In the context of the logistic growth model shown in the second image, what does the horizontal asymptote $$y = C$$ represent?

Accepted Answer

The maximum possible population (carrying capacity)

Question 3

Which of the following best describes the behavior of the function in the third image?

Accepted Answer

An exponential decay function with $$k \u003c 0$$

Question 4

Suppose a population grows according to $$A = 500 e$$^{0.04 t}$$. What is the population at t = 10$?

Accepted Answer

$$A = 500 e$$^{0.4}$$ \approx 745$$

Question 5

A radioactive substance decays according to $$A = A_0$ e$$^{-0.002 t}$$. If A_0$ = 100$$ grams, how much remains after 500 years?

Accepted Answer

$$A = 100 e$$^{-1}$$ \approx 36.8$$ grams

Question 6

Given the logistic growth model $$f(t) = \frac{30000}{1 + 20 e$$^{-1.5 t}$$}$$, what is the limiting population as $$t 	o \infty$$?

Accepted Answer

$$30000$$

Question 7

A cup of coffee is heated to $$90^\circC $$and left in a $$20^\circC $$room. After 10 minutes, it cools to $$60^\circC. $$Using Newton’s Law of Cooling, $$T = 20 + 70 e$$^{kt}$$, what is the value of k$?

Accepted Answer

$$k = \frac{\ln(40/70)}{10}$$

Question 8

If a population doubles every 12 years, what is the exponential growth rate $$k in$$ $$A = A_0$ e$$^{kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{12}$$

Question 9

Which of the following is the correct way to rewrite $$y = 5 	imes 3^x in $$base $$e$$?

Accepted Answer

$$y = 5 e$$^{(\ln 3)x}$$

Question 10

A fossil contains 60% of its original Carbon-14. If the half-life of Carbon-14 is 5730 years, how old is the fossil? (Use $$A = A_0$ e$$^{kt}$$ with k$$ = \frac{\ln(1/2)}{5730}$$)

Accepted Answer

$$t = \frac{\ln(0.6)}{\ln(1/2)/5730} \approx 4230$$ years

Question 11

Which of the following statements is true about the exponential decay function $$A = A_0$ e$$^{kt}$$ when k$$ \u003c 0$$?

Accepted Answer

The function decreases as $$t$$ increases.

Question 12

If $$y = 2.557 	imes 1.017^x$$ models world population in billions, $$x$$ years after 1949, what is the equivalent model in base $$e$$?

Accepted Answer

$$y = 2.557 e$$^{0.017x}$$

Question 13

A scientist finds that a sample has 25% of its original radioactive material. If the decay constant is $$k = -0.000121$$, how long has the sample been decaying?

Accepted Answer

$$t = \frac{\ln(0.25)}{-0.000121} \approx 11,448$$ years

Question 14

In Newton’s Law of Cooling, $$T = C + (T_0$ - C) e$$^{kt}$$, what does C$ represent?

Accepted Answer

The temperature of the surroundings

Question 15

Which of the following best describes the rate of change in a logistic growth model as shown in the second image?

Accepted Answer

The rate of growth increases at first, then decreases as it approaches the limit

Exponential Growth, Decay, and Logistic Models in Precalculus

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