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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 1

Linear Functions

Educators


Problem 1

Find the slope of each line.
Through $(4,5)$ and $(-1,2)$

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Problem 2

Find the slope of each line.
Through $(5,-4)$ and $(1,3)$

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Problem 3

Find the slope of each line.
Through $(8,4)$ and $(8,-7)$

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Problem 4

Find the slope of each line.
Through $(1,5)$ and $(-2,5)$

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Problem 5

Find the slope of each line.
$$
y=x
$$

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Problem 6

Find the slope of each line.
$$
y=3 x-2
$$

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Problem 7

Find the slope of each line.
$$
5 x-9 y=11
$$

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Problem 8

Find the slope of each line.
$$
4 x+7 y=1
$$

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Problem 9

Find the slope of each line.
$$
x=5
$$

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Problem 10

Find the slope of each line.
The $x$ -axis

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Problem 11

Find the slope of each line.
$$
y=8
$$

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Problem 12

Find the slope of each line.
$$
y=-6
$$

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Problem 13

Find the slope of each line.
A line parallel to $6 x-3 y=12$

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Problem 14

Find the slope of each line.
A line perpendicular to $8 x=2 y-5$

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Problem 15

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(1,3), m=-2$

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Problem 16

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(2,4), m=-1$

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Problem 17

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(-5,-7), m=0$

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Problem 18

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(-8,1),$ with undefined slope

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Problem 19

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(4,2)$ and $(1,3)$

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Problem 20

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(8,-1)$ and $(4,3)$

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Problem 21

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(2 / 3,1 / 2)$ and $(1 / 4,-2)$

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Problem 22

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(-2,3 / 4)$ and $(2 / 3,5 / 2)$

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Problem 23

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(-8,4)$ and $(-8,6)$

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Problem 24

In Exercises $15-24,$ find an equation in slope-intercept form for each line.
Through $(-1,3)$ and $(0,3)$

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Problem 25

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
$x$ -intercept $-6, y$ -intercept $-3$

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Problem 26

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
$x$ -intercept $-2, y$ -intercept 4

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Problem 27

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Vertical, through $(-6,5)$

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Problem 28

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Horizontal, through $(8,7)$

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Problem 29

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Through $(-4,6),$ parallel to $3 x+2 y=13$

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Problem 30

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Through $(2,-5),$ parallel to $2 x-y=-4$

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Problem 31

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Through $(3,-4),$ perpendicular to $x+y=4$

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Problem 32

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
Through $(-2,6),$ perpendicular to $2 x-3 y=5$

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Problem 33

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
The line with $y$ -intercept 4 and perpendicular to $x+5 y=7$

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Problem 34

In Exercises $25-34$ , find an equation for each line in the form $a x+b y=c,$ where $a, b,$ and $c$ are integers with no factor common to all three and $a \geq 0$ .
The line with $x$ -intercept $-2 / 3$ and perpendicular to $2 x-y=4$

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Problem 35

Do the points $(4,3),(2,0),$ and $(-18,-12)$ lie on the same line? Explain why or why not. (Hint: Find the slopes between the points.)

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Problem 36

Find $k$ so that the line through $(4,-1)$ and $(k, 2)$ is
a. parallel to $2 x+3 y=6$
b. perpendicular to $5 x-2 y=-1$

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Problem 37

Use slopes to show that the quadrilateral with vertices at $(1,3),$ $(-5 / 2,2),(-7 / 2,4),$ and $(2,1)$ is a parallelogram.

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Problem 38

Use slopes to show that the square with vertices at $(-2,5),$ $(4,5),(4,-1),$ and $(-2,-1)$ has diagonals that are perpendicular.

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Problem 39

For the lines in Exercises 39 and $40,$ which of the following is closest to the slope of the line? (a) 1 (b) 2 (c) 3 (d) 21 (e) 22 (f) $-3$

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Problem 40

For the lines in Exercises 39 and $40,$ which of the following is closest to the slope of the line? (a) 1 (b) 2 (c) 3 (d) 21 (e) 22 (f) $-3$

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Problem 41

In Exercises 41 and $42,$ estimate the slope of the lines.

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Problem 42

In Exercises 41 and $42,$ estimate the slope of the lines.

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Problem 43

To show that two perpendicular lines, neither of which is vertical, have slopes with a product of $-1,$ go through the following steps. Let line $L_{1}$ have equation $y=m_{1} x+b_{1},$ and let $L_{2}$ have equation $y=m_{2} x+b_{2},$ with $m_{1}>0$ and $m_{2}<0$ . Assume that $L_{1}$ and $L_{2}$ are perpendicular, and use right triangle $M P N$ shown in the figure. Prove each of the following statements.
a. $M Q$ has length $m_{1}$
b. $Q N$ has length $-m_{2}$
c. Triangles $M P Q$ and $P N Q$ are similar.
d. $m_{1} / 1=1 /\left(-m_{2}\right)$ and $m_{1} m_{2}=-1$

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Problem 44

Consider the equation $\frac{x}{a}+\frac{y}{b}=1$
a. Show that this equation represents a line by writing it in the form $y=m x+b$
b. Find the $x-$ and $y$ -intercepts of this line.
c. Explain in your own words why the equation in this exercise
is known as the intercept form of a line.

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Problem 45

Graph each equation.
$$
y=x-1
$$

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Problem 46

Graph each equation.
$$
y=4 x+5
$$

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Problem 47

Graph each equation.
$$
y=-4 x+9
$$

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Problem 48

Graph each equation.
$$
y=-6 x+12
$$

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Problem 49

Graph each equation.
$$
2 x-3 y=12
$$

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Problem 50

Graph each equation.
$$
3 x-y=-9
$$

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Problem 51

Graph each equation.
$$
3 y-7 x=-21
$$

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Problem 52

Graph each equation. $5 y+6 x=11$

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Problem 53

Graph each equation.
$$
5 y+6 x=11
$$

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Problem 54

Graph each equation.
$$
x=4
$$

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Problem 55

Graph each equation.
$$
x+5=0
$$

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Problem 56

Graph each equation.
$$
y+8=0
$$

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Problem 57

Graph each equation.
$$
y=2 x
$$

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Problem 58

Graph each equation.
$$
y=-5 x
$$

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Problem 59

Graph each equation.
$$
x+4 y=0
$$

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Problem 60

Graph each equation.
$$
3 x-5 y=0
$$

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Problem 61

Sales The sales of a small company were $\$ 27,000$ in its second year of operation and $\$ 63,000$ in its fifth year. Let $y$ represent sales in the $x$ th year of operation. Assume that the data can be approximated by a straight line.
a. Find the slope of the sales line, and give an equation for the line in the form $y=m x+b .$
b. Use your answer from part a to find out how many years must pass before the sales surpass $\$ 100,000$ .

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Problem 62

Use of Cellular Telephones The following table shows the subscribership of cellular telephones in the United States (in millions) for even-numbered years between 2000 and 2008. Source: Time Almanac 2010.
a. Plot the data by letting correspond to 2000. Discuss how well the data fit a straight line.
b. Determine a linear equation that approximates the number of subscribers using the points (0, 109.48) and (8, 270.33).
c. Repeat part b using the points (2, 140.77) and (8, 270.33). d. Discuss why your answers to parts b and c are similar but not identical.
e. Using your equations from parts b and c, approximate the

number of cellular phone subscribers in the year 2007. Com-
pare your result with the actual value of 255.40 million.

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Problem 63

Consumer Price Index The Consumer Price Index (CPI) is a measure of the change in the cost of goods over time. The index was 100 for the three-year period centered on 1983. For simplicity, we will assume that the CPI was exactly 100 in 1983. Then the CPI of 215.3 in 2008 indicates that an item that cost $1.00 in 1983 would cost $2.15 in 2008. The CPI has been increasing approximately linearly over the last few decades. Source: Time Almanac 2010.
a. Use this information to determine an equation for the CPI in terms of , which represents the years since 1980.
b. Based on the answer to part a, what was the predicted value of the CPI in 2000? Compare this estimate with the actual CPI of 172.2.
c. Describe the rate at which the annual CPI is changing.

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Problem 64

HIV Infection The time interval between a person’s initial infection with HIV and that person’s eventual development of AIDS symptoms is an important issue. The method of infection with HIV affects the time interval before AIDS develops. One study of HIV patients who were infected by intravenous drug use found that 17% of the patients had AIDS after 4 years, and 33% had developed the disease after 7 years.
The relationship between the time interval and the percentage of patients with AIDS can be modeled accurately with a linear equation. Source: Epidemiologic Review.
a. Write a linear equation $y=m t+b$ that models this data, using the ordered pairs $(4,0.17)$ and $(7,0.33) .$
b. Use your equation from part a to predict the number of years before half of these patients will have AIDS.

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Problem 65

Exercise Heart Rate To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minute) should be in the target heart rate zone. The lower limit of this zone is found by taking 70% of the difference between 220 and your age. The upper limit is found by using 85%. Source: Physical Fitness.
a. Find formulas for the upper and lower limits $(u \text { and } l)$ as lin- ear equations involving the age $x .$
b. What is the target heart rate zone for a 20-year-old?
c. What is the target heart rate zone for a 40-year-old?
d. Two women in an aerobics class stop to take their pulse and are surprised to find that they have the same pulse. One woman is 36 years older than the other and is working at the upper limit of her target heart rate zone. The younger woman is working at the lower limit of her target heart rate zone. What are the ages of the two women, and what is their pulse?
e. Run for 10 minutes, take your pulse, and see if it is in your tar- get heart rate zone. (After all, this is listed as an exercise!)

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Problem 66

Ponies Trotting A 1991 study found that the peak vertical force on a trotting Shetland pony increased linearly with the pony’s speed, and that when the force reached a critical level, the pony switched from a trot to a gallop. For one pony, the critical force was 1.16 times its body weight. It experienced a force of 0.75 times its body weight at a speed of 2 meters per second and a force of 0.93 times its body weight at 3 meters per second. At what speed did the pony switch from a trot to a gallop? Source: Science.

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Problem 67

Life Expectancy Some scientists believe there is a limit to how long humans can live. One supporting argument is that during the last century, life expectancy from age 65 has increased more slowly than life expectancy from birth, so eventually these two will be equal, at which point, according to these scientists, life expectancy should increase no further. In 1900, life expectancy at birth was 46 yr, and life expectancy at age 65 was 76 yr. In 2004, these figures had risen to 77.8 and 83.7, respectively. In
both cases, the increase in life expectancy has been linear. Using these assumptions and the data given, find the maximum life expectancy for humans. Source: Science.

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Problem 68

Child Mortality Rate The mortality rate for children under 5 years of age around the world has been declining in a roughly linear fashion in recent years. The rate per 1000 live births was 90 in 1990 and 65 in 2008. Source: World Health Organization.
a. Determine a linear equation that approximates the mortality rate in terms of time , where represents the number of years since 1900.
b. If this trend continues, in what year will the mortality rate first drop to 50 or below per 1000 live births?

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Problem 69

Health Insurance The percentage of adults in the United States without health insurance increased at a roughly linear rate from 1999, when it was 17.2%, to 2008, when it was 20.3%. Source: The New York Times.
a. Determine a linear equation that approximates the percentage of adults in the United States without health insurance in terms of time $t,$ where $t$ represents the number of years since 1990 .
b. If this trend were to continue, in what year would the percent- age of adults without health insurance be at least 25$\%$ ?

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Problem 70

Marriage The following table lists the U.S. median age at first marriage for men and women. The age at which both groups marry for the first time seems to be increasing at a roughly linear rate in recent decades. Let t correspond to the number of years since 1980. Source: U.S. Census Bureau.
table can't copy
a. Find a linear equation that approximates the data for men, using the data for the years 1980 and 2005.
b. Repeat part a using the data for women.
c. Which group seems to have the faster increase in median age at first marriage?
d. In what year will the men’s median age at first marriage reach 30?
e. When the men’s median age at first marriage is 30, what will the median age be for women?

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Problem 71

Immigration In 1950, there were 249,187 immigrants admitted to the United States. In 2008, the number was 1,107,126. Source: 2008 Yearbook of Immigration Statistics.
a. Assuming that the change in immigration is linear, write an equation expressing the number of immigrants, , in terms of $t$ the number of years after 1900.
b. Use your result in part a to predict the number of immigrants admitted to the United States in 2015.
c. Considering the value of the -intercept in your answer to part a, discuss the validity of using this equation to model the number of immigrants throughout the entire 20th century.

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Problem 72

Global Warming $\operatorname{In} 1990$ , the Intergovernmental Panel on Climate Change predicted that the average temperature on Earth would rise $0.3^{\circ} \mathrm{C}$ per decade in the absence of international controls on greenhouse emissions. Let $t$ measure the time in years since $1970,$ when the average global temperature was $15^{\circ} \mathrm{C}$ . Source: Science News.
a. Find a linear equation giving the average global temperature in degrees Celsius in terms of $t,$ the number of years since 1970 .
b. Scientists have estimated that the sea level will rise by 65 $\mathrm{cm}$ if the average global temperature rises to $19^{\circ} \mathrm{C}$ . According to your answer to part a, when would this occur?

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Problem 73

Galactic Distance The table lists the distances (in megaparsecs where 1 megaparsec $\approx 3.1 \times 10^{19} \mathrm{km}$ ) and velocities (in kilometers per second) of four galaxies moving rapidly away from Earth. Source: Astronomical Methods and Calculations, and Fundamental Astronomy.
$$
\begin{array}{lll}{\text { Galaxy }} & {\text { Distance }} & {\text { Velocity }} \\ \hline \text { Virga } & {15} & {1600} \\ {\text { Ursa Minor }} & {200} & {15,000} \\ {\text { Corona Borealis }} & {290} & {24,000} \\ {\text { Bootes }} & {520} & {40,000}\end{array}
$$
a. Plot the data points letting x represent distance and y represent velocity. Do the points lie in an approximately linear pattern?
b. Write a linear equation to model this data, using the ordered pair
c. The galaxy Hydra has a velocity of 60,000 km per sec. Use your equation to approximate how far away it is from Earth.
d. The value of m in the equation is called the Hubble constant. The Hubble constant can be used to estimate the age of the universe A (in years) using the formula
$$
A=\frac{9.5 \times 10^{11}}{m}
$$
Approximate $A$ using your value of $m$

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Problem 74

News/Talk Radio From 2001 to 2007, the number of stations carrying news/talk radio increased at a roughly linear rate, from 1139 in 2001 to 1370 in 2007. Source: State of the Media.
a. Find a linear equation expressing the number of stations carrying news/talk radio, , in terms of the years since 2000.
b. Use your answer from part a to predict the number of stations carrying news/talk radio in 2008. Compare with the actual number of 2046. Discuss how the linear trend from 2001 to 2007 might have changed in 2008.

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Problem 75

Tuition The table lists the annual cost (in dollars) of tuition and fees at private four-year colleges for selected years. (See Example 14.) Source: The College Board.
$$
\begin{array}{ll}{\text { Year }} & {\text { Tuition and Fees }} \\ \hline 200 & {16,072} \\ \hline 2002 & {16,072} \\ {2004} & {20,045} \\ {2006} & {25,308} \\ {2008} & {25,177} \\ {2009} & {26,273}\end{array}
$$
a. Sketch a graph of the data. Do the data appear to lie roughly along a straight line?
b. Let $t=0$ correspond to the year 2000 . Use the points $(0,16,072)$ and $(9,26,273)$ to determine a linear equation that models the data. What does the slope of the graph of the equation indicate?
e. Discuss the accuracy of using this equation to estimate the cost of private college in 2025 .

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