Q: A certain bacteria culture doubles every 20 minutes. If you start with 1 cell, how long will it take for the culture to reach the mass of the Earth, given that the mass of the Earth is $$6 \times 10$$^{27}$$ grams and the mass of one E. coli cell is 10$$^{-12}$$ grams?

$$t = 20 \log_2$ (6 \times 10$$^{39}$$)$$

Q: A population $$P$$ grows at a yearly rate equal to $$0.3$$ times the population. If $$P(0) = 100,000$$, what is the population after 4 years?

$$P(4) = 100,000 e$$^{0.3 \times 4}$$

Q: Which of the following is the correct formula for the amount remaining after $$t$$ half-lives for a substance with initial amount $$A_0$$?

$$A(t) = A_0$ \left( \frac{1}{2} \$$right)^{t/\text{half-life}$$}$$

Q: A scientist observes that a population of bacteria triples every hour. If the initial population is 100, what is the population after 5 hours?

$$100 \times 3$$^{5}$$ = 24,300$$

Question 1

A certain bacteria culture doubles every 20 minutes. If you start with 1 cell, how long will it take for the culture to reach the mass of the Earth, given that the mass of the Earth is $$6 	imes 10$$^{27}$$ grams and the mass of one E. coli cell is 10$$^{-12}$$ grams?

Accepted Answer

$$t = 20 \log_2$ (6 	imes 10$$^{39}$$)$$

Question 2

Given the exponential growth model $$m(t) = (1) 2$$^{t/20}$$, what is the derivative m$$'(t)$$ with respect to t$?

Accepted Answer

$$m'(t) = (1) 2$$^{t/20}$$ (\ln 2) \frac{1}{20}$$

Question 3

An isotope has a half-life of 10 hours. If you start with 12 grams, what is the amount left after 30 hours?

Accepted Answer

$$12 \left( \frac{1}{2} \$$right)^{3}$$ = 1.5$$ grams

Question 4

Given the differential equation $$\frac{dy}{dt} = ky$$ with $$k \u003e 0$$, which of the following is the general solution?

Accepted Answer

$$y(t) = Ce$$^{kt}$$

Question 5

A population $$P$$ grows at a yearly rate equal to $$0.3$$ times the population. If $$P(0) = 100,000$$, what is the population after 4 years?

Accepted Answer

$$P(4) = 100,000 e$$^{0.3 	imes 4}$$

Question 6

Which of the following best describes the meaning of a 'half-life' in the context of exponential decay?

Accepted Answer

The time required for a quantity to decrease to half its initial value.

Question 7

Given the differential equation $$y'' - x y' = 0$$, which of the following is a solution?

Accepted Answer

$$y = x^2$

Question 8

If $$y(t)$$ satisfies $$\frac{dy}{dt} = -ky$$ with $$k \u003e 0$$ and $$y(0) = y_0$, what is the solution?

Accepted Answer

$$y(t) = y_0$ e$$^{-kt}$$

Question 9

A radioactive substance decays according to $$m(t) = m_0$ \left( \frac{1}{2} \$$right)^{t/10}$$. What is the rate of decay at t = 6$ hours if m_0$ = 12$$ grams?

Accepted Answer

$$m'(6) = 12 \left( \frac{1}{2} \$$right)^{6/10}$$ \ln \left( \frac{1}{2} \right) \frac{1}{10}$$

Question 10

Which of the following statements about initial value problems is correct?

Accepted Answer

An initial value problem specifies a unique solution to a differential equation by providing the value of the function at a specific time.

Question 11

Suppose $$y(t)$$ satisfies $$\frac{dy}{dt} = ky$$ and $$y(0) = y_0$. What is the value of the constant C$ in the general solution y(t$$) = $$Ce^{kt}$$?

Accepted Answer

$$C = y_0$

Question 12

If a population grows according to $$P(t) = P_0$ e$$^{rt}$$, what does the parameter r$ represent?

Accepted Answer

The relative (per capita) growth rate of the population.

Question 13

Which of the following is the correct formula for the amount remaining after $$t$$ half-lives for a substance with initial amount $$A_0$$?

Accepted Answer

$$A(t) = A_0$ \left( \frac{1}{2} \$$right)^{t/	ext{half-life}$$}$$

Question 14

Given $$y(t) = Ce$$^{kt}$$, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$\frac{dy}{dt} = $$kCe^{kt}$$

Question 15

A scientist observes that a population of bacteria triples every hour. If the initial population is 100, what is the population after 5 hours?

Accepted Answer

$$100 	imes 3$$^{5}$$ = 24,300$$

Question 16

Which of the following best describes the solution to the differential equation $$\frac{dy}{dt} = ky$$ with $$k \u003c 0$$?

Accepted Answer

The solution represents exponential decay.

Exponential Growth, Decay Models, and Differential Equations

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