Q: A biologist is studying a bacteria population that doubles every 3 hours. If the initial population is 500 bacteria, which formula best models the population after $$t$$ hours?

$$P(t) = 500 \times 2$$^{t/3}$$

Question 1

A biologist is studying a bacteria population that doubles every 3 hours. If the initial population is 500 bacteria, which formula best models the population after $$t$$ hours?

Accepted Answer

$$P(t) = 500 	imes 2$$^{t/3}$$

Question 2

A radioactive substance has a half-life of 8 years. If you start with 200 grams, how much remains after 24 years?

Accepted Answer

$$25$$ grams

Question 3

Which of the following is the correct formula for the amount $$A in an $$account after $$t$$ years with continuous compounding at annual rate $$r$$ and principal $$P$$?

Accepted Answer

$$A = P e$$^{rt}$$

Question 4

A population of fish in a lake is modeled by $$P(t) = 1200 e$$^{0.04t}$$, where t$ is in years. What is the relative growth rate?

Accepted Answer

$$0.04$$

Question 5

If you invest $$\5000$ at an annual interest rate of 6$$\%$$ compounded quarterly, what is the value of n$ in the periodic compound interest formula?

Accepted Answer

$$n = 4$$

Question 6

Which step is necessary to solve the exponential equation $$3^x = 20$$ for $$x$$?

Accepted Answer

Take the logarithm of both sides

Question 7

A city’s population is decreasing at a rate proportional to its size. If the initial population is $$P_0$$ and the relative decay rate is $$k$$, which formula models the population after $$t$$ years?

Accepted Answer

$$P(t) = P_0$ e$$^{-kt}$$

Question 8

To convert a decimal to a percent, which operation should you perform?

Accepted Answer

Multiply by $$100$$

Question 9

A sample of a radioactive isotope decays according to $$A(t) = 80 e$$^{-0.2t}$$. What is the half-life of the isotope?

Accepted Answer

$$t = \frac{\ln 2}{0.2}$$

Question 10

If $$A = P \left(1 + \frac{r}{n}\$$right)^{nt}$$, what does r$ represent in the context of compound interest?

Accepted Answer

The annual interest rate as a decimal

Question 11

A scientist measures a substance and finds it decays at a rate of $$5\%$$ per year. Which is the correct exponential decay model for the amount $$A$$ after $$t$$ years, starting with $$A_0$$?

Accepted Answer

$$A(t) = A_0$ e$$^{-0.05t}$$

Question 12

Which of the following best describes the meaning of 'relative growth rate' in the exponential growth model $$P(t) = P_0$ e$$^{kt}$$?

Accepted Answer

The constant $$k$$ representing the proportional rate of growth per unit time

Question 13

A savings account offers $$3\%$$ annual interest compounded monthly. If you deposit $$\2000$, what is the value of r$ and n$ in the periodic compound interest formula?

Accepted Answer

$$r = 0.03$$, $$n = 12$$

Question 14

Which of the following is NOT a correct step when solving an exponential equation of the form $$b^x = c$$?

Accepted Answer

Multiply both sides by $$x$$

Question 15

A population of rabbits is modeled by $$P(t) = 300 e$$^{0.07t}$$. How long will it take for the population to triple?

Accepted Answer

$$t = \frac{\ln 3}{0.07}$$

Question 16

If you have $$\1000$ in an account earning 4$$\%$ interest compounded annually, how much will you have after 5 years?

Accepted Answer

$$1000 	imes (1 + 0.04)^5$

Question 17

A scientist finds that a sample of carbon-14 has decayed to $$25\% of $$its original amount. If the half-life of carbon-14 is 5730 years, how many half-lives have passed?

Accepted Answer

2

Question 18

Which formula would you use to model the exponential decay of a medication in the bloodstream, given initial amount $$A_0$$ and decay rate $$k$$?

Accepted Answer

$$A(t) = A_0$ e$$^{-kt}$$

Question 19

A population of bacteria is modeled by $$P(t) = 1000 e$$^{0.05t}$$. What is the population after 10 hours?

Accepted Answer

$$1000 e$$^{0.5}$$

Applications of Exponential and Logarithmic Functions

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