Question 1

Which investment option will pay the most interest?

Accepted Answer

A) 10.2% compounded quarterly 
B) 10.4% compounded semiannually 
C) 10.0% compounded continuously 
D) These investments all pay the same amount of interest. 
E) 10.6% compounded annually 
A) 10.2% compounded quarterly 
B) 10.4% compounded semiannually 
C) 10.0% compounded continuously 
D) These investments all pay the same amount of interest. 
E) 10.6% compounded annually B

Question 2

The value V (in millions of dollars)of a famous painting can be modeled by $V = 10 e ^ {k t }$ where t represents the year,with t = 0 corresponding to 2000.In 2008,the same painting was sold for $65 million.Predict the value of the painting in 2018.(Round your answer to two decimal places. ) ​

Accepted Answer

A) $674.61 million 
B) $874.61 million 
C) $474.61 million 
D) $774.61 million 
E) $574.61 million 
A) $674.61 million 
B) $874.61 million 
C) $474.61 million 
D) $774.61 million 
E) $574.61 million A

Question 3

Find the magnitude R of an earthquake of intensity $I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }$ .​ $I = 16000$ ​

Accepted Answer

A) 3.20 
B) 5.20 
C) 4.20 
D) 2.20 
E) 6.20 
A) 3.20 
B) 5.20 
C) 4.20 
D) 2.20 
E) 6.20 C

Question 4

Determine the principal that must be invested at rate r,compounded monthly,so that $\$ 900,000$ will be available for retirement in t years.​ $r = 3 \frac { 1 } { 2 } \% , t = 15$ ​

Accepted Answer

A)  $\$ 532,806.81$ 
B)  $\$ 632,806.81$ 
C)  $\$ 432,806.81$ 
D)  $\$ 732,806.81$ 
E)  $\$ 332,806.81$ 
A)  $\$ 532,806.81$ 
B)  $\$ 632,806.81$ 
C)  $\$ 432,806.81$ 
D)  $\$ 732,806.81$ 
E)  $\$ 332,806.81$

Question 5

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.What is $\left[ \mathrm { H } ^ { + } \right]$ if the $\mathrm { pH } = 3.8$ ?

Accepted Answer

A) 3.800 
B)  $6.31 \times 10 ^ { - 4 }$ 
C)  $6.31 \times 10 ^ { - 3 }$ 
D)  $1.58 \times 10 ^ { - 3 }$ 
E)  $1.58 \times 10 ^ { - 4 }$ 
A) 3.800 
B)  $6.31 \times 10 ^ { - 4 }$ 
C)  $6.31 \times 10 ^ { - 3 }$ 
D)  $1.58 \times 10 ^ { - 3 }$ 
E)  $1.58 \times 10 ^ { - 4 }$

Question 6

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ ,where $\left[ \mathrm { H } ^ { + } \right]$ is the hydrogen ion concentration in the solution.If a sample of rain has a pH of 3.3,how many times higher is its $\left[ \mathrm { H } ^ { + } \right]$ than pure water's,which has a pH of 7?

Accepted Answer

A)  $5.0 \times 10 ^ { 4 }$ 
B)  $2.0 \times 10 ^ { 4 }$ 
C)  $2.0 \times 10 ^ { 3 }$ 
D)  $5.0 \times 10 ^ { 3 }$ 
E) 7 
A)  $5.0 \times 10 ^ { 4 }$ 
B)  $2.0 \times 10 ^ { 4 }$ 
C)  $2.0 \times 10 ^ { 3 }$ 
D)  $5.0 \times 10 ^ { 3 }$ 
E) 7

Question 7

Complete the table for the radioactive isotope.Round your answer to two decimal places. $$\begin{array} { | c | c | c | c | } 
\hline \text { Isotope } & \begin{array} { c } 
\text { Half-life } \
\text { (years) }
\end{array} & \text { Initial Quantity } & \begin{array} { c } 
\text { Amount after 1000 } \
\text { years }
\end{array} \
\hline { } ^ { 239 } \mathrm { Pu } & 24,100 & 2.4 g & --\
\hline
\end{array}$$

Accepted Answer

A) Amount after 1000 years: $4.33 \mathrm {~g}$ 
B) Amount after 1000 years: $5.33 \mathrm {~g}$ 
C) Amount after 1000 years: $3.33 \mathrm {~g}$ 
D) Amount after 1000 years: $6.33 \mathrm {~g}$ 
E) Amount after 1000 years: $2.33 \mathrm {~g}$ 
A) Amount after 1000 years: $4.33 \mathrm {~g}$ 
B) Amount after 1000 years: $5.33 \mathrm {~g}$ 
C) Amount after 1000 years: $3.33 \mathrm {~g}$ 
D) Amount after 1000 years: $6.33 \mathrm {~g}$ 
E) Amount after 1000 years: $2.33 \mathrm {~g}$

Question 8

The population P of a culture of bacteria is described by the equation $P = 1300 e ^ { 0.052 t }$ ,where t is the time,in hours,relative to the time at which the population was 1300.What was the population at $t = 3$ hours? ​

Accepted Answer

A)  $P ( 3 ) \approx 1419$ 
B)  $P ( 3 ) \approx 1519$ 
C)  $P ( 3 ) \approx 1319$ 
D)  $P ( 3 ) \approx 1619$ 
E)  $P ( 3 ) \approx 1719$ 
A)  $P ( 3 ) \approx 1419$ 
B)  $P ( 3 ) \approx 1519$ 
C)  $P ( 3 ) \approx 1319$ 
D)  $P ( 3 ) \approx 1619$ 
E)  $P ( 3 ) \approx 1719$

Question 9

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } 
\hline \text { Initial investment } & \text { Annual rate } & \text { Time to double } & \text { Amount after 10 years } \
\hline \ldots& 6.93 \% & \ldots & \$ 3999.41 \
\hline
\end{array}$$

Accepted Answer

A) Initial investment: $\$ 2,000$ Time to double: 10 yr 
B) Initial investment: $\$ 1,950$ Time to double: 10 yr 
C) Initial investment: $\$ 2,200$ Time to double: 10 yr 
D) Initial investment: $\$ 2,050$ Time to double: 10 yr 
E) Initial investment: $\$ 2,100$ Time to double: 10 yr 
A) Initial investment: $\$ 2,000$ Time to double: 10 yr 
B) Initial investment: $\$ 1,950$ Time to double: 10 yr 
C) Initial investment: $\$ 2,200$ Time to double: 10 yr 
D) Initial investment: $\$ 2,050$ Time to double: 10 yr 
E) Initial investment: $\$ 2,100$ Time to double: 10 yr

Question 10

The populations P (in thousands)of Pittsburgh,Pennsylvania from 2000 through 2007 can be modeled by $P = \frac { 2632 } { 1 + 0.083 e ^ { 0.0500 t } }$ where t represents the year,with $t = 0$ corresponding to 2000.Use the model to find the numbers of cell sites in the year 2009.
​

Accepted Answer

A) 2,326,853.00 
B) 2,327,853.00 
C) 2,329,853.00 
D) 2,328,853.00 
E) 2,330,853.00 
A) 2,326,853.00 
B) 2,327,853.00 
C) 2,329,853.00 
D) 2,328,853.00 
E) 2,330,853.00

Question 11

An initial investment of $4000 grows at an annual interest rate of 5% compounded continuously.How long will it take to double the investment?

Accepted Answer

A) 1 year 
B) 14.40 years 
C) 13.86 years 
D) 14.86 years 
E) 13.40 years 
A) 1 year 
B) 14.40 years 
C) 13.86 years 
D) 14.86 years 
E) 13.40 years

Question 12

If $1 is invested in an account over a 10-year period,the amount in the account,where t represents the time in years,is given by $$A = 1 + 0.075 ||t|| \text { or } A = e ^ { 0.07 t }$$ depending on whether the account pays simple interest at $$7 \frac { 1 } { 2 } \%$$ or continuous compound interest at 7%.Graph each function on the same set of axes.Which grows at a higher rate? ​

Accepted Answer

A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   ​ 
D) ​   
E) ​

Question 13

The population P (in thousands)of Orlando,Florida from 2000 through 2007 can be modeled by $P = 1530.6 e ^ { k t }$ where t represents the year,with $t = 0$ corresponding to 2000.In 2006,the population of Orlando,Florida was about 1,883,000.00.Find the value of k. ​

Accepted Answer

A) k = 0.03183 
B) k = 0.02863 
C) k = 0.02163 
D) k = 0.03453 
E) k = 0.03063 
A) k = 0.03183 
B) k = 0.02863 
C) k = 0.02163 
D) k = 0.03453 
E) k = 0.03063

Question 14

Select the correct graph for the given function​ $$y = \ln ( x + 2 )$$ ​

Accepted Answer

A) ​   
B) ​   
C) ​   
D) ​   
E) ​   
A) ​   
B) ​   
C) ​   
D) ​   
E) ​

Question 15

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table. $$\begin{array} { | l | l | l | } 
\hline x & 0 & 4 \
\hline y & 5 & 1 \
\hline
\end{array}$$

Accepted Answer

A)  $y = 5 e ^ { - 0.4024 x }$ 
B)  $y = 5 e ^ { 0.5024 x }$ 
C)  $y = - \frac { 1 } { 5 } e ^ { 0.4024 x }$ 
D)  $y = \frac { 1 } { 5 } e ^ { - 0.6024 x }$ 
E)  $y = - \frac { 1 } { 5 } e ^ { - 0.4024 x }$ 
A)  $y = 5 e ^ { - 0.4024 x }$ 
B)  $y = 5 e ^ { 0.5024 x }$ 
C)  $y = - \frac { 1 } { 5 } e ^ { 0.4024 x }$ 
D)  $y = \frac { 1 } { 5 } e ^ { - 0.6024 x }$ 
E)  $y = - \frac { 1 } { 5 } e ^ { - 0.4024 x }$

Question 16

Find the intensity I of an earthquake measuring R on the Richter scale $\text { (let } I _ { 0 } = 1 \text { ) }$ . ​
Costa Rica in 2009, $R = 5.70$ ​

Accepted Answer

A)  $10 ^ { 5.70 } \approx 503,187$ 
B)  $10 ^ { 5.70 } \approx 501,187$ 
C)  $10 ^ { 5.70 } \approx 506,187$ 
D)  $10 ^ { 5.70 } \approx 496,187$ 
E)  $10 ^ { 5.70 } \approx 499,187$ 
A)  $10 ^ { 5.70 } \approx 503,187$ 
B)  $10 ^ { 5.70 } \approx 501,187$ 
C)  $10 ^ { 5.70 } \approx 506,187$ 
D)  $10 ^ { 5.70 } \approx 496,187$ 
E)  $10 ^ { 5.70 } \approx 499,187$

Question 17

The number y of hits a new search-engine website receives each month can be modeled by $y = 4080 e ^ { k t }$ where t represents the number of months the website has been operating.In the website's third month,there were 10,000 hits.Find the value of k,and use this value to predict the number of hits the website will receive after 22 months. ​

Accepted Answer

A)  $k = 0.2988$ ;About 184,894,691,979 hits 
B)  $k = 0.2988$ ;About 2,921,047 hits 
C)  $k = 0.2988$ ;About 22,832,533,585 hits 
D)  $k = 0.2988$ ;About 57,970,933 hits 
E)  $k = 0.2988$ ;About 1,150,488,266 hits 
A)  $k = 0.2988$ ;About 184,894,691,979 hits 
B)  $k = 0.2988$ ;About 2,921,047 hits 
C)  $k = 0.2988$ ;About 22,832,533,585 hits 
D)  $k = 0.2988$ ;About 57,970,933 hits 
E)  $k = 0.2988$ ;About 1,150,488,266 hits

Question 18

Carbon dating presumes that,as long as a plant or animal is alive,the proportion of its carbon that is 14C is constant.The amount of 14C in an object made from harvested plants,like paper,will decline exponentially according to the equation $A = A _ { 0 } e ^ { - 0.0001213 t }$ ,where A represents the amount of 14C in the object,Ao represents the amount of 14C in living organisms,and t is the time in years since the plant was harvested.If an archeological artifact has 35% as much 14C as a living organism,how old would you predict it to be? Round to the nearest year.

Accepted Answer

A) 5715 years 
B) 29,310 years 
C) 93 years 
D) 8655 years 
E) 15,427 years 
A) 5715 years 
B) 29,310 years 
C) 93 years 
D) 8655 years 
E) 15,427 years

Question 19

Complete the table for a savings account in which interest is compounded continuously.$$\begin{array} { | c | c | c | c | } 
\hline \text { Initial investment } & \text { Annual } \% \text { rate } & \text { Time to double } & \text { Amount after 10 years } \
\hline  { \$ 500 } & 9 \frac { 1 } { 2 } \% & ....& .....\\
\hline
\end{array}$$
(Round the answer up to two decimal places. )

Accepted Answer

A) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1292.85$ 
B) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1542.85$ 
C) Time to double: $2.30 y r$ Amount after 10 years: $\$ 1292.85$ 
D) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1792.85$ 
E) Time to double: $12.30 \mathrm { yr }$ Amount after 10 years: $\$ 1292.85$ 
A) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1292.85$ 
B) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1542.85$ 
C) Time to double: $2.30 y r$ Amount after 10 years: $\$ 1292.85$ 
D) Time to double: $7.30 y r$ Amount after 10 years: $\$ 1792.85$ 
E) Time to double: $12.30 \mathrm { yr }$ Amount after 10 years: $\$ 1292.85$

Question 20

Complete the table for the radioactive isotope. $$\begin{array} { | c | c | c | c | } 
\hline \text { Isotope } & \begin{array} { c } 
\text { Half-life } \
\text { (years) }
\end{array} & \text { Initial Quantity } & \begin{array} { c } 
\text { Amount after 1000 } \
\text { years }
\end{array} \
\hline { } ^ { 226 } \mathrm { Ra } & 1599& --& 4 \mathrm {~g} \
\hline
\end{array}$$

Accepted Answer

A) Initial Quantity: $8.17 g$ 
B) Initial Quantity: $10.17 \mathrm {~g}$ 
C) Initial Quantity: $6.17 g$ 
D) Initial Quantity: $9.17 \mathrm {~g}$ 
E) Initial Quantity: $7.17 \mathrm {~g}$ 
A) Initial Quantity: $8.17 g$ 
B) Initial Quantity: $10.17 \mathrm {~g}$ 
C) Initial Quantity: $6.17 g$ 
D) Initial Quantity: $9.17 \mathrm {~g}$ 
E) Initial Quantity: $7.17 \mathrm {~g}$

Exam 22: Exponential and Logarithmic Models

Which investment option will pay the most interest?

Find the magnitude R of an earthquake of intensity I( let I0=1 ) I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }I( let I0​=1 ) .​ I=16000I = 16000I=16000 ​

Determine the principal that must be invested at rate r,compounded monthly,so that $900,000\$ 900,000$900,000 will be available for retirement in t years.​ r=312%,t=15r = 3 \frac { 1 } { 2 } \% , t = 15r=321​%,t=15 ​

Complete the table for the radioactive isotope.Round your answer to two decimal places. Isotope Half-life (years) Initial Quantity Amount after 1000 years 24,100 2.4g --

The population P of a culture of bacteria is described by the equation P=1300e0.052tP = 1300 e ^ { 0.052 t }P=1300e0.052t ,where t is the time,in hours,relative to the time at which the population was 1300.What was the population at t=3t = 3t=3 hours? ​

Complete the table for a savings account in which interest is compounded continuously. Initial investment Annual rate Time to double Amount after 10 years \ldots 6.93\% \ldots \ 3999.41

An initial investment of $4000 grows at an annual interest rate of 5% compounded continuously.How long will it take to double the investment?

The population P (in thousands)of Orlando,Florida from 2000 through 2007 can be modeled by P=1530.6ektP = 1530.6 e ^ { k t }P=1530.6ekt where t represents the year,with t=0t = 0t=0 corresponding to 2000.In 2006,the population of Orlando,Florida was about 1,883,000.00.Find the value of k. ​

Select the correct graph for the given function​ y=ln⁡(x+2)y = \ln ( x + 2 )y=ln(x+2) ​

Find the exponential model y=aebxy = a e ^ { b x }y=aebx that fits the points shown in the table. x 0 4 y 5 1

Find the intensity I of an earthquake measuring R on the Richter scale (let I0=1 ) \text { (let } I _ { 0 } = 1 \text { ) } (let I0​=1 ) . ​ Costa Rica in 2009, R=5.70R = 5.70R=5.70 ​

Complete the table for a savings account in which interest is compounded continuously. Initial investment Annual \% rate Time to double Amount after 10 years \ 500 9\% .... ..... (Round the answer up to two decimal places. )

Complete the table for the radioactive isotope. Isotope Half-life (years) Initial Quantity Amount after 1000 years 1599 -- 4

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Parametric Equations

Polar Coordinates

Polar Equations of Conics

Graphs of Polar Equations

Find the magnitude R of an earthquake of intensity $I \left( \text { let } I _ { 0 } = 1 \right. \text { ) }$ . $I = 16000$

Determine the principal that must be invested at rate r,compounded monthly,so that $\$ 900,000$ will be available for retirement in t years. $r = 3 \frac { 1 } { 2 } \% , t = 15$

The population P of a culture of bacteria is described by the equation $P = 1300 e ^ { 0.052 t }$ ,where t is the time,in hours,relative to the time at which the population was 1300.What was the population at $t = 3$ hours?

The population P (in thousands)of Orlando,Florida from 2000 through 2007 can be modeled by $P = 1530.6 e ^ { k t }$ where t represents the year,with $t = 0$ corresponding to 2000.In 2006,the population of Orlando,Florida was about 1,883,000.00.Find the value of k.

Select the correct graph for the given function $y = \ln ( x + 2 )$

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table. x 0 4 y 5 1

Find the intensity I of an earthquake measuring R on the Richter scale $\text { (let } I _ { 0 } = 1 \text { ) }$ . Costa Rica in 2009, $R = 5.70$