Deck 22: Exponential and Logarithmic Models

Question 1

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

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

Answer 1

13.86 years

Question 2

Select the correct graph for the given function​ $$y = 7 e ^ { - x / 4 }$$ ​

A)​
B)​
C)​
D)​
E)​

Answer 2

​

Question 3

An initial investment of $5000 doubles in value in 6.3 years.Assuming continuous compounding,what was the interest rate? Round to the nearest tenth of a percent.

A)11.0%
B)4.8%
C)5.5%
D)6.3%
E)100%

Answer 3

11.0%

Question 4

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. )

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 5

Select the correct graph for the given function​ $$y = 5 e ^ { - ( x - 2 ) ^ { 2 } / 5 }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 6

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual \% rate } & \text { Time to double } & \text { Amount after 10 years } \\\hline { \$ 750 } & ....& 9 \frac { 3 } { 4 } y r &.... \\\hline\end{array}$$
(Round the answer up to two decimal places. )

A)Annual rate: $7.11 \%$ Amount after 10 years: $\$ 1527.02$
B)Annual rate: $11.11 \%$ Amount after 10 years: $\$ 1527.02$
C)Annual rate: $7.11 \%$ Amount after 10 years: $\$ 1027.02$
D)Annual rate: $7.11 \%$ Amount after 10 years: $\text { \$2027.02 }$
E)Annual rate: $9.11 \%$ Amount after 10 years: $\$ 1527.02$

Question 7

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual rate } & \text { Time to double } & \text { Amount after 10 years } \\\hline \$ 11,000 & \ldots& \ldots & \$ 21996.76 \\\hline\end{array}$$ (Round the answer up to two decimal places. )

A)Annual rate: $8.93 \%$ Time to double: 10 $y r$
B)Annual rate: $6.93 \%$ Time to double: 15 yr
C)Annual rate: $10.93 \%$ Time to double: 10 $y r$
D)Annual rate: $6.93 \%$ Time to double: 14 yr
E)Annual rate: $6.93 \%$ Time to double: 10 $y r$

Question 8

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual rate } & \text { Time to double } & \text { Amount after 10 years } \\\hline \$7,000& \ldots& \ldots & \$ 13143.27 \\\hline\end{array}$$ (Round the answer up to two decimal places. )

A)Annual rate: $6.3 \%$ Time to double: 14 yr
B)Annual rate: $6.3 \%$ Time to double: 11 yr
C)Annual rate: $10.3 \%$ Time to double: 11 yr
D)Annual rate: $8.3 \%$ Time to double: 11 yr
E)Annual rate: $6.3 \%$ Time to double: 15 yr

Question 9

Which investment option will pay the most interest?

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

Question 10

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual rate } & \text { Time to double } & \text { Amount after 10 years } \\\hline ....&5.78 \% & .... & \$ 10694.82 \\\hline\end{array}$$ (Round the answer up to two decimal places. )

A)Initial investment: $\$ 10000$ Time to double: 13 yr
B)Initial investment: $\$ 9000$ Time to double: 10 yr
C)Initial investment: $\$ 7000$ Time to double: 12 yr
D)Initial investment: $\$ 6,000$ Time to double: 12 yr
E)Initial investment: $\$ 8000$ Time to double: 12 yr

Question 11

Select the correct graph for the given function​ $$y = \frac { 6 } { 1 + e ^ { - 2 x } }$$ ​

A)​
B)​ ​
C)​
D)​
E)​

Question 12

Select a scatter plot of the given data. ​
R $7 \%$ $8 \%$ $11 \%$ $12 \%$ $14 \%$ $15 \%$ t $16.24$ $14.27$ $10.53$ $9.69$ $8.38$ $7.86$ ​
​
​

A)​
B)​
C)​
D)​
E)​

Question 13

​Complete the table for the time t (in years)necessary for P dollars to triple if interest is compounded continuously at rate r. ​​
$$\begin{array} { | l | l | l | l | l | l | l | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & & & & & & \\\hline\end{array}$$ (Round the answer up to two decimal places. )​

A)​​ $$\begin{array} { | l | c | c | c | c | c | c | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & 7.32 & 7.85 & 8.45 & 9.99 & 10.99 & 13.73 \\\hline\end{array}$$
B)​​ $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & 13.73 & 9.99 & 10.99 & 8.45 & 7.32 & 7.85 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & 13.73 & 10.99 & 9.99 & 7.85 & 7.32 & 8.45 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & 7.32 & 7.85 & 9.99 & 8.45 & 10.99 & 13.73 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 8 \% & 10 \% & 11 \% & 13 \% & 14 \% & 15 \% \\\hline t & 13.73 & 10.99 & 9.99 & 8.45 & 7.85 & 7.32 \\\hline\end{array}$$

Question 14

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

A)​
B)​
C)​
D)​
E)​

Question 15

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual rate } & \text { Time to double } & \begin{array} { c } \text { Amount after 10 } \\\text { years }\end{array} \\\hline \$ 9,000 & \ldots & 13 \mathrm { yr } & \cdots \\\hline\end{array}$$
(Round the answer up to two decimal places. )

A)Annual rate: $5.33 \%$ Amount after 10 years: $\$ 14,836.33$
B)Annual rate: $5.33 \%$ Amount after 10 years: $\$ 15,836.33$
C)Annual rate: $5.33 \%$ Amount after 10 years: $\$ 15336.33$
D)Annual rate: $9.33 \%$ Amount after 10 years: $\$ 15336.33$
E)Annual rate: $7.33 \%$ Amount after 10 years: $\$ 15336.33$

Question 16

Select the correct graph for the given function ​​ $$y = 5 + \log ( x + 2 )$$ ​

A)​ ​
B)​
C)​
D)​
E)​

Question 17

Complete the table for a savings account in which interest is compounded continuously. $$\begin{array} { | c | c | c | c | } \hline \text { Initial investment } & \text { Annual \% rate } & \text { Time to double } & \begin{array} { c } \text { Amount after 10 } \\\text { years }\end{array} \\\hline \$ 5000 & 5.5 \% & \cdots & \cdots \\\hline\end{array}$$
(Round the answer up to two decimal places. )

A)Time to double: $13 yr$ Amount after 10 years: $$\$ 9166.27$$
B)Time to double: $$13 yr$$ Amount after 10 years: $\$ 9666.27$
C)Time to double: $18 yr$ Amount after 10 years: $\$ 8666.27$
D)Time to double: $8 y r$ Amount after 10 years: $\$ 8666.27$
E)Time to double: $13 yr$ Amount after 10 years: $\$ 8666.27$

Question 18

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

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

Question 19

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

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 20

Select the correct graph for the given function​ $$y = 4 e ^ { x / 2 }$$ ​

A)​
B)​
C)​
D)​
E)​

Question 21

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? ​

A)​
B)​
C)​ ​
D)​
E)​

Question 22

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

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: $7.17 \mathrm {~g}$

Question 23

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

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 24

Сomplete the table for the radioactive isotope. $$\begin{array} { | c | c | c | c | } \hline \text { Isotope } & \begin{array} { c } \text { Half-life } \\\text { (years) }\end{array} & \text { Initial Quantity } & \begin{array} { c } \text { Amount after } 1000 \\\text { years }\end{array} \\\hline { } ^ { 14 } \mathrm { C } & 5715 & --& 2 \mathrm {~g} \\\hline\end{array}$$

A)Initial Quantity: $2.26 \mathrm {~g}$
B)Initial Quantity: $3.26 \mathrm {~g}$
C)Initial Quantity: $5.26 \mathrm {~g}$
D)Initial Quantity: $4.26 \mathrm {~g}$
E)Initial Quantity: $6.26 \mathrm {~g}$

Question 25

Find the exponential model $$y = a e ^ { 0.5756 x }$$ that fits the points shown in the graph.​ ​

A) $$y = \frac { 1 } { 2 } e ^ { - 0.5756 x }$$
B) $y = - \frac { 1 } { 2 } x e ^ { - 0.5756 }$
C) $y = \frac { 1 } { 2 } x e ^ { 0.5756 }$
D)​​ $y = \frac { 1 } { 2 } e ^ { 0.5756 x }$
E) $y = - \frac { 1 } { 2 } e ^ { - 0.5756 x }$

Question 26

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. ) ​

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

Question 27

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? ​

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 28

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

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 29

Complete the table for the radioactive isotope. $\begin{array}{|l|l|l|l|}\hline \text { Isotope } & \begin{array}{l}\text { Half-life } \\\text { (years) }\end{array} & \text { Initial Quantity } & \begin{array}{l}\text { Amount after } 1000 \\\text { years }\end{array} \\\hline{ }^{239} \mathrm{Pu} &24,100 &-- & 0.3g\\\hline\end{array}$

A)Initial Quantity: $1.31 \mathrm {~g}$
B)Initial Quantity: $3.31 \mathrm {~g}$
C)Initial Quantity: $4.31 \mathrm {~g}$
D)Initial Quantity: $2.31 \mathrm {~g}$
E)Initial Quantity: $0.31 \mathrm {~g}$

Question 30

The population P of a bacteria culture is modeled by $P = 4100 e ^ { k t }$ ,where t is the time in hours.If the population of the culture was 5800 after 40 hours,how long does it take for the population to double? Round to the nearest tenth of an hour.

A)54.8 hours
B)8.9 hours
C)79.9 hours
D)81.7 hours
E)56.6 hours

Question 31

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. ​

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 32

The populations P (in thousands)of Horry County,South Carolina from 1970 through 2007 can be modeled by $P = - 13.5 + 92.2 e ^ { 0.0282 t }$ where t represents the year,with $t = 0$ corresponding to 1970.Use the model to complete the table.Round your answer to two decimal places. $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & & & & & \\\hline\end{array}$$

A) $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & 78.70 & 108.74 & 148.56 & 201.35 & 248.24 \\\hline\end{array}$$
B) $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & 78.70 & 118.74 & 148.56 & 258.24 & 201.35 \\\hline\end{array}$$
C) $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & 78.70 & 158.56 & 148.56 & 201.35 & 248.24 \\\hline\end{array}$$
D) $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & 78.70 & 211.35 & 158.56 & 201.35 & 248.24 \\\hline\end{array}$$
E) $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\\hline \text { Population } & 78.70 & 258.24 & 258.24 & 211.35 & 148.56 \\\hline\end{array}$$

Question 33

Find the exponential model $y = a e ^ { 0.7675 x }$ that fits the points shown in the graph.​ ​

A) $$y = e ^ { - 0.7675 x }$$
B) $y = e ^ { 0.7675 x }$
C) $$y = - e ^ { - 0.7675 x }$$
D) $y = x e ^ { 0.7675 }$
E) $y = - x e ^ { - 0.7675 }$

Question 34

Complete the table for the radioactive isotope.Round your answer to two decimal places. $$\begin{array} { | c | c | c | c | } \hline \text { Isotope } & \begin{array} { c } \text { Half-life } \\\text { (years) }\end{array} & \text { Initial Quantity } & \begin{array} { c } \text { Amount after 1000 } \\\text { years }\end{array} \\\hline { } ^ { 226 } \mathrm { Ra } & 1599 & 5 g & --\\\hline\end{array}$$

A)Amount after 1000 years: $5.24 \mathrm {~g}$
B)Amount after 1000 years: $7.24 \mathrm {~g}$
C)Amount after 1000 years: $3.24 \mathrm {~g}$
D)Amount after 1000 years: $6.24 g$
E)Amount after 1000 years: $4.24 \mathrm {~g}$

Question 35

Find the exponential model $y = a e ^ { b x }$ that fits the points shown in the table. $$\begin{array} { | c | c | c | } \hline x & 0 & 3 \\\hline y & 6 & 1.5 \\\hline\end{array}$$

A) $y = \frac { 1 } { 2 } e ^ { 0.6621 x }$
B) $y = e ^ { - 0.5621 x }$
C) $y = - e ^ { 0.4621 x }$
D) $y = 6 e ^ { - 0.4621 x }$
E) $y = 2 e ^ { 0.4621 x }$

Question 36

Select a scatter plot of the given data. r
4%
6%
8%
10%
12%
14%
T
28)01
18)85
14)27
11)53
9)69
8)38
​

A)​
B)​
C)​
D)​ ​
E)​

Question 37

The population P (in thousands)of Reno,Nevada from 2000 through 2007 can be modeled by $P = 346.8 e ^ { k t }$ where t represents the year,with $t = 0$ corresponding to 2000.In 2005,the population of Reno was about 395,000.According to the model,during what year will the population reach 486,000.00? ​

A)2013
B)2005
C)2021
D)2017
E)2009

Question 38

Complete the table for the radioactive isotope.Round your answer to two decimal places. $\begin{array}{|l|l|l|l|}\hline \text { Isotope } & \begin{array}{l}\text { Half-life } \\\text { (years) }\end{array} & \text { Initial Quantity } & \text { Amount after } 1000 \\\hline{ }^{14} \mathrm{C} & 5715& 5 g & --\\\hline\end{array}$

A)Amount after 1000 years: $5.43 \mathrm {~g}$
B)Amount after 1000 years: $6.43 \mathrm {~g}$
C)Amount after 1000 years: $4.43 \mathrm {~g}$
D)Amount after 1000 years: $7.43 \mathrm {~g}$
E)Amount after 1000 years: $8.43 \mathrm {~g}$

Question 39

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. ​

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

Question 40

​Complete the table for the time t (in years)necessary for P dollars to triple if interest is compounded annually at rate r.Round your answer to two decimal places. ​​
$$\begin{array} { | l | l | l | l | l | l | l | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & & & & & & \\\hline\end{array}$$

A) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & 11.53 & 9.69 & 8.38 & 7.40 & 6.64 & 6.03 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & 6.03 & 6.64 & 7.40 & 8.38 & 9.69 & 11.53 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & 6.03 & 6.64 & 8.38 & 7.40 & 9.69 & 11.53 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & 11.53 & 8.38 & 9.69 & 7.40 & 6.03 & 6.64 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | c | c | c | c | c | } \hline r & 10 \% & 12 \% & 14 \% & 16 \% & 18 \% & 20 \% \\\hline t & 11.53 & 9.69 & 8.38 & 6.64 & 6.03 & 7.40 \\\hline\end{array}$$

Question 41

An initial investment of $2000 grows at an annual interest rate of 4% compounded continuously.How long will it take to double the investment?

A)17.33 years
B)18.33 years
C)18.00 years
D)17.00 years
E)1 year

Question 42

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 $\mathrm { pH } = 3.8$ ?

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 43

The population P of a culture of bacteria is described by the equation ,where t is the time,in hours,relative to the time at which the population was 1600.
(a)What was the population at hours? Show your work.
(b)After how many hours will the population reach 10000.00? Round to the nearest tenth of an hour.Show your work.

Question 44

A population growing at an annual rate r will triple in a time t given by the formula $t = \frac { \ln 3 } { r }$ .If the growth rate remains constant and equals 9% per year,how long will it take the population of the town to triple? ​

A)2.2 years
B)5.3 years
C)6.6 years
D)1 years
E)12.2 years

Question 45

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?

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 46

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

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 47

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 3.4,how many times higher is its $\left[ \mathrm { H } ^ { + } \right]$ than pure water's,which has a pH of 7?

A) $4.0 \times 10 ^ { 3 }$
B) $2.5 \times 10 ^ { 4 }$
C)7
D) $2.5 \times 10 ^ { 3 }$
E) $4.0 \times 10 ^ { 4 }$

Question 48

Tritium,a radioactive isotope of hydrogen,has a half-life of 12.4 years.Of an initial sample of 33 grams,how much will remain after 69 years? ​

A)0 grams
B)29.1351 grams
C)10.9074 grams
D)8.2500 grams
E)0.6973 grams

Question 49

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,A_o 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 30% as much ¹⁴C as a living organism,how old would you predict it to be? Round to the nearest year.

A)28,040 years
B)5715 years
C)9926 years
D)14,758 years
E)100 years

Question 50

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones.The numbers of cell sites from 1985 through 2008 can be modeled by $y = \frac { 237,101 } { 1 + 1950 e ^ { - 0.355 t } }$ where t represents the year,with $t = 5$ corresponding to 1985.Use the model to find the numbers of cell sites in the year 2007 . ​

A)211,071
B)209,071
C)208,071
D)207,071
E)210,071

Question 51

Find the magnitude R of an earthquake of intensity $\left. I \text { (let } I _ { 0 } = 1 \right)$ .​ $I = 270300000$ ​

A)7.43
B)6.43
C)8.43
D)10.43
E)9.43

Question 52

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

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

Question 53

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,A_o 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.

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

Question 54

Find the intensity I of an earthquake measuring R on the Richter scale $\left( \text { let } I _ { 0 } = 1 \right)$ . ​
Southern Sumatra,Indonesia in 2007, $R = 8.25$ ​

A) $10 ^ { 8.25 } \approx 177,822,941$
B) $10 ^ { 8.25 } \approx 177,827,941$
C) $10 ^ { 8.25 } \approx 177,825,941$
D) $10 ^ { 8.25 } \approx 177,832,941$
E) $10 ^ { 8.25 } \approx 177,829,941$

Question 55

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.
​

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 56

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 $\mathrm { pH } = 6.8$ ?

A) $1.58 \times 10 ^ { - 7 }$
B) $6.31 \times 10 ^ { - 7 }$
C) $6.31 \times 10 ^ { - 6 }$
D) $1.58 \times 10 ^ { - 6 }$
E)6.800