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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