Q: Suppose a bacteria culture doubles every 20 minutes. If you start with 1 cell, how long will it take for the culture to reach $$6 \times 10$$^{39}$$ cells (the approximate number needed to equal the mass of the Earth)?

$$t = 20 \log_2$ (6 \times 10$$^{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?

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

Question 1

Suppose a bacteria culture doubles every 20 minutes. If you start with 1 cell, how long will it take for the culture to reach $$6 	imes 10$$^{39}$$ cells (the approximate number needed to equal the mass of the Earth)?

Accepted Answer

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

Question 2

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

Accepted Answer

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

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 best describes the solution?

Accepted Answer

$$y(t) = Ce$$^{kt}$$, where C$ is a constant

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

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

Question 6

Which of the following differential equations models exponential decay?

Accepted Answer

$$\frac{dy}{dt} = -ky$$, $$k \u003e 0$$

Question 7

If $$y(t)$$ satisfies $$\frac{dy}{dt} = 0.12y$$ and $$y(0) = 12$$, what is the explicit solution for $$y(t)$$?

Accepted Answer

$$y(t) = 12e$$^{0.12t}$$

Question 8

A radioactive substance decays according to $$m(t) = m_0$ \left( \frac{1}{2} \$$right)^{t/h}$$, where h$ is the half-life. What is the rate of decay at t = h$?

Accepted Answer

$$m'(h) = m_0$ \left( \frac{1}{2} \right) \ln \left( \frac{1}{2} \right) \frac{1}{h}$$

Question 9

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 10

If a population decays at a rate proportional to its size, which of the following best models the population at time $$t$$?

Accepted Answer

$$y(t) = Ce$$^{-kt}$$, \quad k \u003e 0$

Question 11

A scientist observes that a chemical's mass halves every 8 hours. If the initial mass is 40 grams, how much remains after 24 hours?

Accepted Answer

$$40 \left( \frac{1}{2} \$$right)^{3}$$ = 5$$ grams

Question 12

Which of the following is the general solution to the differential equation $$\frac{dy}{dt} = ky$$?

Accepted Answer

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

Question 13

If $$y(t) = Ce$$^{kt}$$ and y(0$$) = $$y_0$$, what is the value of $$C$$?

Accepted Answer

$$C = y_0$

Question 14

A population $$P$$ grows according to $$P(t) = 100,000e$$^{0.3t}$$. What is the instantaneous rate of growth at t = 2$?

Accepted Answer

$$P'(2) = 100,000e$$^{0.6}$$ 	imes 0.3$$

Question 15

Which of the following best describes the meaning of the half-life of a substance?

Accepted Answer

The time required for half of the substance to decay.

Question 16

Given $$y'(t) = ky(t)$$ and $$y(0) = y_0$, what is the unique solution?

Accepted Answer

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

Question 17

If a population grows at a rate proportional to its size, which of the following is true about the solution to the differential equation?

Accepted Answer

The solution is an exponential function.

Question 18

A scientist starts with 20 grams of a radioactive isotope with a half-life of 5 hours. How much remains after 15 hours?

Accepted Answer

$$20 \left( \frac{1}{2} \$$right)^{3}$$ = 2.5$$ grams

Exponential Growth, Decay Models, and Differential Equations

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