Precalculus with Limits (2010)

Ron Larson, David C. Falvo

Chapter 10

Topics in Analytic Geometry

Educators


Problem 1

The ________ of a nonhorizontal line is the positive angle (less than measured counterclockwise from the axis to the line.

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Problem 2

If a nonvertical line has inclination $\theta$ and slope $m,$ then $m=$________.

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Problem 3

If two nonperpendicular lines have slopes $m_{1}$ and $m_{2},$ the angle between the two lines is $\tan \theta=$___________.

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Problem 4

The distance between the point $\left(x_{1}, y_{1}\right)$ and the line $A x+B y+C=0$ is given by $d=$________.

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Problem 5

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
GRAPH CANNOT COPY

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Problem 6

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
GRAPH CANNOT COPY

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Problem 7

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
GRAPH CANNOT COPY

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Problem 8

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
GRAPH CANNOT COPY

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Problem 9

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
$$\theta=\frac{\pi}{3}\text{radians} $$

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Problem 10

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
$$\theta=\frac{5 \pi}{6}\text{radians} $$

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Problem 11

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
$$\theta=\text{1.27 radians} $$

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Problem 12

In Exercises $5-12,$ find the slope of the line with inclination $\theta$
$$\theta=\text{2.88 radians} $$

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Problem 13

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=-1$$

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Problem 14

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=-2$$

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Problem 15

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=1$$

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Problem 16

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=2$$

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Problem 17

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=\frac{3}{4}$$

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Problem 18

In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$
$$m=-\frac{5}{2}$$

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Problem 19

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(\sqrt{3}, 2),(0,1)$$

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Problem 20

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(1,2 \sqrt{3}),(0, \sqrt{3})$$

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Problem 21

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(-\sqrt{3},-1),(0,-2)$$

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Problem 22

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(3, \sqrt{3}),(6,-2 \sqrt{3})$$

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Problem 23

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(6,1),(10,8)$$

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Problem 24

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(12,8),(-4,-3)$$

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Problem 25

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(-2,20),(10,0)$$

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Problem 26

In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$$(0,100),(50,0)$$

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Problem 27

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$2 x+2 y-5=0$$

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Problem 28

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$x-\sqrt{3} y+1=0$$

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Problem 29

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$3 x-3 y+1=0$$

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Problem 30

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$\sqrt{3} x-y+2=0$$

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Problem 31

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$x+\sqrt{3} y+2=0$$

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Problem 32

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$-2 \sqrt{3} x-2 y=0$$

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Problem 33

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$6 x-2 y+8=0$$

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Problem 34

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$4 x+5 y-9=0$$

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Problem 35

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$5 x+3 y=0$$

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Problem 36

In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.
$$2 x-6 y-12=0$$

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Problem 37

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{aligned} 3 x+y &=3 \\ x-y &=2 \end{aligned}$$

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Problem 38

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{x+3 y=2} \\ {x-2 y=-3}\end{array}$$

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Problem 39

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{aligned} x-y &=0 \\ 3 x-2 y &=-1 \end{aligned}$$

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Problem 40

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{2 x-y=2} \\ {4 x+3 y=24}\end{array}$$

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Problem 41

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{aligned} x-2 y &=7 \\ 6 x+2 y &=5 \end{aligned}$$

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Problem 42

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{5 x+2 y=16} \\ {3 x-5 y=-1}\end{array}$$

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Problem 43

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{x+2 y=8} \\ {x-2 y=2}\end{array}$$

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Problem 44

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{3 x-5 y=3} \\ {3 x+5 y=12}\end{array}$$

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Problem 45

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{0.05 x-0.03 y=0.21} \\ {0.07 x+0.02 y=0.16}\end{array}$$

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Problem 46

In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.
$$\begin{array}{l}{0.02 x-0.05 y=-0.19} \\ {0.03 x+0.04 y=0.52}\end{array}$$

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Problem 47

ANGLE MEASUREMENT In Exercises $47-50$ , find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.
GRAPH CANNOT COPY

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Problem 48

ANGLE MEASUREMENT In Exercises $47-50$ , find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.
GRAPH CANNOT COPY

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Problem 49

ANGLE MEASUREMENT In Exercises $47-50$ , find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.
GRAPH CANNOT COPY

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Problem 50

ANGLE MEASUREMENT In Exercises $47-50$ , find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.
GRAPH CANNOT COPY

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Problem 51

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{cc}{\text {Point}} & {\text {Line}} \\ {(0,0)} & {4 x+3 y=0}\end{array}$$

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Problem 52

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{cc}{\text {Point}} & {\text {Line}} \\ {(0,0)} & {2 x-y=4}\end{array}$$

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Problem 53

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(2,3)} & {3 x+y=1}\end{array}$$

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Problem 54

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(-2,1)} & {x-y=2}\end{array}$$

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Problem 55

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(-6,2)} & {x+1=0}\end{array}$$

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Problem 56

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(2,1)} & {-2 x+y-2=0}\end{array}$$

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Problem 57

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(0,8)} & {6 x-y=0}\end{array}$$

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Problem 58

In Exercises $51-58$ , find the distance between the point and the line.
$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(4,2)} & {x-y=20}\end{array}$$

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Problem 59

In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.
$$A=(0,0), B=(1,4), C=(4,0)$$

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Problem 60

In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.
$$A=(0,0), B=(4,5), C=(5,-2)$$

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Problem 61

In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.
$$A=\left(-\frac{1}{2}, \frac{1}{2}\right), B=(2,3), C=\left(\frac{5}{2}, 0\right)$$

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Problem 62

In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.
$$A=(-4,-5), B=(3,10), C=(6,12)$$

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Problem 63

In Exercises 63 and $64,$ find the distance between the parallel lines.
$$\begin{array}{l}{x+y=1} \\ {x+y=5}\end{array}$$

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Problem 64

In Exercises 63 and $64,$ find the distance between the parallel lines.
$$\begin{array}{l}{3 x-4 y=1} \\ {3 x-4 y=10}\end{array}$$

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Problem 65

ROAD GRADE A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

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Problem 66

ROAD GRADE A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

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Problem 67

PITCH OF A ROOF A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof.

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Problem 68

CONVEYOR DESIGN A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel.
(a) Draw a diagram that gives a visual representation of the problem.
(b) Find the inclination of the conveyor.
(c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

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Problem 69

TRUSS Find the angles and shown in the drawing of the roof truss.

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Problem 70

The Falls Incline Railway in Niagara Falls, Ontario, Canada is an inclined railway that was designed to carry people from the City of Niagara Falls to Queen Victoria Park. The railway is approximately 170 feet long with a 36% uphill grade (see figure).
(a) Find the inclination $\theta$ of the railway.
(b) Find the change in elevation from the base to the top of the railway.
(c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track.
(d) Sketch a graph of the equation you found in part (c).

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Problem 71

TRUE OR FALSE? In Exercises 71 and $72,$ determine whether the statement is true or false. Justify your answer.
A line that has an inclination greater than $\pi / 2$ radians has a negative slope.

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Problem 72

TRUE OR FALSE? In Exercises 71 and $72,$ determine whether the statement is true or false. Justify your answer.
To find the angle between two lines whose angles of inclination $\theta_{1}$ and $\theta_{2}$ are known, substitute $\theta_{1}$ and $\theta_{2}$ for $m_{1}$ and $m_{2},$ respectively, in the formula for the angle between two lines.

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Problem 73

Consider a line with slope $m$ and $y$ -intercept $(0,4)$ .
(a) Write the distance $d$ between the origin and the line as a function of $m .$
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the origin and the line.
(d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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Problem 74

CAPSTONE Discuss why the inclination of a line can be an angle that is larger than $\pi / 2,$ but the angle between two lines cannot be larger than $\pi / 2 .$ Decide whether the following statement is true or false: "The inclination of a line is the angle between the line and the $x$ -axis." Explain.

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Problem 75

Consider a line with slope $m$ and $y$ -intercept $(0,4)$ .
(a) Write the distance $d$ between the point $(3,1)$ and the line as a function of $m .$
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the point and the line.
(d) Is it possible for the distance to be 0 ? If so, what is the slope of the line that yields a distance of 0$?$
(e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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