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Linear Equations: Homework

Module by: UniqU, LLC. E-mail the author

Based on: Applied Finite Mathematics: Chapter 01 by Rupinder Sekhon

Summary: This chapter covers principles of linear equations. After completing this chapter students should be able to: graph a linear equation; find the slope of a line; determine an equation of a line; solve linear systems; and complete application problems using linear equations.

GRAPHING A LINEAR EQUATION

Work the following problems.

Exercise 1

Is the point (2, 3) on the line 5x−2y=45x−2y=4 size 12{5x - 2y=4} {}?

Solution

Yes

Exercise 2

Is the point (1, – 2) on the line 6x−y=46x−y=4 size 12{6x - y=4} {}?

Exercise 3

For the line 3x−y=123x−y=12 size 12{3x - y=12} {}, complete the following ordered pairs.

(2, ) ( , 6)

(0, ) ( , 0)

Solution

(2, –6) (6, 6)

(0,–12) (4, 0)

Exercise 4

For the line 4x+3y=244x+3y=24 size 12{4x+3y="24"} {}, complete the following ordered pairs.

(3, ) ( , 4)

(0, ) ( , 0)

Exercise 5

Graph y=2x+3y=2x+3 size 12{y=2x+3} {}

Solution

Figure 1
The line y=2x+3 passing through the points (-1,1) and (0,3) on a Cartesian graph.

Exercise 6

Graph y=−3x+5y=−3x+5 size 12{y= - 3x+5} {}

Exercise 7

Graph y=4x−3y=4x−3 size 12{y=4x - 3} {}

Solution

Figure 2
The line y=4x-3 passing through the points (0,-3) and (2,5) on a Cartesian graph.

Exercise 8

Graph x−2y=8x−2y=8 size 12{x - 2y=8} {}

Exercise 9

Graph 2x+y=42x+y=4 size 12{2x+y=4} {}

Solution

Figure 3
The line 2x+y=4 passing through the point (0,4) and (2,0) on a Cartesian graph.

Exercise 10

Graph 2x−3y=62x−3y=6 size 12{2x - 3y=6} {}

Exercise 11

Graph 2x+4=02x+4=0 size 12{2x+4=0} {}

Solution

Figure 4
A line passing through the point (-2,0) on a Cartesian graph.

Exercise 12

Graph 2y−6=02y−6=0 size 12{2y - 6=0} {}

Exercise 13

Graph the following three equations on the same set of coordinate axes.

y=x+1   y=2x+1  y=−x+1y=x+1   size 12{y=x+1}      {}y=2x+1   size 12{y=2x+1}      {}y=−x+1 size 12{y= - x+1} {}
(1)

Solution

Figure 5
The lines y=-x+1, y=2x+7, and y=x+1 intersecting at the point (0,1) on a Cartesian graph.

Exercise 14

Graph the following three equations on the same set of coordinate axes.

y=2x+1  y=2x  y=2x−1y=2x+1   size 12{y=2x+1} {}y=2x   size 12{y=2x} {}y=2x−1 size 12{y=2x - 1} {}
(2)

Exercise 15

Graph the line using the parametric equations

x=1+2t, y=3+tx=1+2t size 12{x=1+2t} {},y=3+t size 12{y=3+t} {}
(3)

Solution

Figure 6
A line passing through the points (-5,0), (1,3), and (3,4) on a Cartesian graph.

Exercise 16

Graph the line using the parametric equations

x=2−3t,   y=1+2tx=2−3t size 12{x=2 - 3t} {},  y=1+2t size 12{y=1+2t} {}
(4)

SLOPE OF A LINE

Find the slope of the line passing through the following pair of points.

Exercise 17

(2, 3) and (5, 9)

Solution

m=2m=2 size 12{m=2} {}
(5)

Exercise 18

(4, 1) and (2, 5)

Exercise 19

(– 1, 1) and (1, 3)

Solution

m=1m=1 size 12{m=1} {}
(6)

Exercise 20

(4, 3) and (– 1, 3)

Exercise 21

(6, – 5) and (4, – 1)

Solution

m=−2m=−2 size 12{m= - 2} {}
(7)

Exercise 22

(5, 3) and (– 1, – 4)

Exercise 23

(3, 4) and (3, 7)

Solution

m=undefinedm=undefined size 12{m="undefined"} {}
(8)

Exercise 24

(– 2, 4) and (– 3, – 2)

Exercise 25

(– 3, – 5) and (– 1, – 7)

Solution

m=−1m=−1 size 12{m= - 1} {}
(9)

Exercise 26

(0, 4) and (3, 0)

Determine the slope of the line from the given equation of the line.

Exercise 27

y=−2x+1y=−2x+1 size 12{y= - 2x+1} {}
(10)

Solution

m=−2m=−2 size 12{m= - 2} {}
(11)

Exercise 28

y=3x−2y=3x−2 size 12{y=3x - 2} {}

Exercise 29

2x−y=62x−y=6 size 12{2x - y=6} {}
(12)

Solution

m=2m=2 size 12{m=2} {}
(13)

Exercise 30

x+3y=6x+3y=6 size 12{x+3y=6} {}
(14)

Exercise 31

3x−4y=123x−4y=12 size 12{3x - 4y="12"} {}
(15)

Solution

m=3/4m=3/4 size 12{m=3/4} {}
(16)

Exercise 32

What is the slope of the x-axis? How about the y-axis?

Graph the line that passes through the given point and has the given slope.

Exercise 33

(1, 2) and m=−3/4m=−3/4 size 12{m= - 3/4} {}

Solution

Figure 7
A line passing through the points (1,2) and (5,-1) on a Cartesian graph.

Exercise 34

(2, –1) and m=2/3m=2/3 size 12{m=2/3} {}

Exercise 35

(0, 2) and m=−2m=−2 size 12{m= - 2} {}

Solution

Figure 8
A line passing through the points (0,2) and (1,0) on a Cartesian graph.

Exercise 36

(2, 3) and m=0m=0 size 12{m=0} {}

DETERMINING THE EQUATION OF A LINE

Write an equation of the line satisfying the following conditions. Write the equation in the form y=mx+by=mx+b size 12{y= ital "mx"+b} {}.

Exercise 37

It passes through the point (3, 10) and has slope=2slope=2 size 12{"slope"=2} {}.

Solution

y=2x+4y=2x+4 size 12{y=2x+4} {}
(17)

Exercise 38

It passes through the point (4,5) and has m=0m=0 size 12{m=0} {}.

Exercise 39

It passes through (3, 5) and (2, – 1).

Solution

y=6x−13y=6x−13 size 12{y=6x - "13"} {}
(18)

Exercise 40

It has slope 3, and its y-intercept equals 2.

Exercise 41

It passes through (5, – 2) and m=2/5m=2/5 size 12{m=2/5} {}.

Solution

y=2/5x−4y=2/5x−4 size 12{y=2/5x - 4} {}
(19)

Exercise 42

It passes through (– 5, – 3) and (10, 0).

Exercise 43

It passes through (4, – 4) and (5, 3).

Solution

y=7x−32y=7x−32 size 12{y=7x - "32"} {}
(20)

Exercise 44

It passes through (7, – 2) and its y-intercept is 5.

Exercise 45

It passes through (2, – 5) and its x-intercept is 4.

Solution

y=5/2x−10y=5/2x−10 size 12{y=5/2x - "10"} {}
(21)

Exercise 46

Its a horizontal line through the point (2, – 1).

Exercise 47

It passes through (5, – 4) and (1, – 4).

Solution

y=−4y=−4 size 12{y= - 4} {}
(22)

Exercise 48

It is a vertical line through the point (3, – 2).

Exercise 49

It passes through (3, – 4) and (3, 4).

Solution

x=3x=3 size 12{x=3} {}
(23)

Exercise 50

It has x-intercept=3x-intercept=3 size 12{x"-intercept"=3} {} and y-intercept=4y-intercept=4 size 12{y"-intercept"=4} {}.

Write an equation of the line satisfying the following conditions. Write the equation in the form Ax+By=CAx+By=C size 12{ ital "Ax"+ ital "By"=C} {}.

Exercise 51

It passes through (3, – 1) and m=2m=2 size 12{m=2} {}.

Solution

2x−y=72x−y=7 size 12{2x - y=7} {}
(24)

Exercise 52

It passes through (– 2, 1) and m=−3/2m=−3/2 size 12{m= - 3/2} {}.

Exercise 53

It passes through (– 4, – 2) and m=3/4m=3/4 size 12{m=3/4} {}.

Solution

3x−4y=−43x−4y=−4 size 12{3x - 4y= - 4} {}
(25)

Exercise 54

Its x-intercept equals 3, and m=−5/3m=−5/3 size 12{m= - 5/3} {}.

Exercise 55

It passes through (2, – 3) and (5, 1).

Solution

4x−3y=174x−3y=17 size 12{4x - 3y="17"} {}
(26)

Exercise 56

It passes through (1, – 3) and (– 5, 5).

APPLICATIONS

In the following application problems, assume a linear relationship holds.

Exercise 57

The variable cost to manufacture a product is $25, and the fixed costs are $1200. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

Solution

y=25x+1200y=25x+1200 size 12{y="25"x+"1200"} {}
(27)

Exercise 58

It costs $90 to rent a car driven 100 miles and $140 for one driven 200 miles. If xx size 12{x} {} is the number of miles driven and yy size 12{y} {} the total cost of the rental, write the cost function.

Exercise 59

The variable cost to manufacture an item is $20, and it costs a total of $750 to produce 20 items. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

Solution

y=20x+350y=20x+350 size 12{y="20"x+"350"} {}
(28)

Exercise 60

To manufacture 30 items, it costs $2700, and to manufacture 50 items, it costs $3200. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

Exercise 61

To manufacture 100 items, it costs $32,000, and to manufacture 200 items, it costs $40,000. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

Solution

y=80x+24000y=80x+24000 size 12{y="80"x+"24000"} {}
(29)

Exercise 62

It costs $1900 to manufacture 60 items, and the fixed costs are $700. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

Exercise 63

A person who weighs 150 pounds has 60 pounds of muscles, and a person that weighs 180 pounds has 72 pounds of muscles. If xx size 12{x} {} represents the body weight and yy size 12{y} {} the muscle weight, write an equation describing their relationship. Use this relationship to determine the muscle weight of a person that weighs 170 pounds.

Solution

y=2/5x;68y=2/5x size 12{y=2/5x} {};68

Exercise 64

A spring on a door stretches 6 inches if a force of 30 pounds is applied, and it stretches 10 inches if a force of 50 pounds is applied. If xx size 12{x} {} represents the number of inches stretched, and yy size 12{y} {} the force applied, write an equation describing the relationship. Use this relationship to determine the amount of force required to stretch the spring 12 inches.

Exercise 65

A male college student who is 64 inches tall weighs 110 pounds, and another student who is 74 inches tall weighs 180 pounds. Assuming the relationship between male students' heights xx size 12{ left (x right )} {}, and weights yy size 12{ left (y right )} {} is linear, write a function to express weights in terms of heights, and use this function to predict the weight of a student who is 68 inches tall.

Solution

y=7x−338y=7x−338 size 12{y = 7x - "338"} {} ; 138

Exercise 66

EZ Clean company has determined that if it spends $30,000 on advertisement, it can hope to sell 12,000 of its Minivacs a year, but if it spends $50,000, it can sell 16,000. Write an equation that gives a relationship between the number of dollars spent on advertisement xx size 12{ left (x right )} {} and the number of minivacs sold yy size 12{ left (y right )} {}.

Exercise 67

The freezing temperatures for Celsius and Fahrenheit scales are 0 degree and 32 degrees, respectively. The boiling temperatures for Celsius and Fahrenheit are 100 degrees and 212 degrees, respectively. Let CC size 12{C} {} denote the temperature in Celsius and FF size 12{F} {} in Fahrenheit. Write the conversion function from Celsius to Fahrenheit, and use this function to convert 25 degrees Celsius into an equivalent Fahrenheit measure.

Solution

F=9/5C+32F=9/5C+32 size 12{F=9/5C+"32"} {}; 77ºF

Exercise 68

By reversing the coordinates in the previous problem, find a conversion function that converts Fahrenheit into Celsius, and use this conversion function to convert 72 degrees Fahrenheit into an equivalent Celsius measure.

Exercise 69

The population of California in the year 1960 was 17 million, and in 1995 it was 32 million. Write the population function, and use this function to find the population of California in the year 2010. (Hint: Use the year 1960 as the base year, that is, assume 1960 as the year zero. This will make 1995, and 2010 as the years 35, and 50, respectively.)

Solution

y=3/7x+17;38.43y=3/7x+17 size 12{y=3/7x+"17"} {};38.43

Exercise 70

In the U. S. the number of people infected with the HIV virus in 1985 was 1,000, and in 1995 that number became 350,000. If the increase in the number is linear, write an equation that will give the number of people infected in any year. If this trend continues, what will the number be in 2010? (Hint: See previous problem.)

Exercise 71

In 1975, an average house in San Jose cost $45,000 and the same house in 1995 costs $195,000. Write an equation that will give the price of a house in any year, and use this equation to predict the price of a similar house in the year 2010.

Solution

y=7500x+45000y=7500x+45000 size 12{y="7500"x+"45000"} {}; $307,500

Exercise 72

An average math text book cost $25 in 1980, and $60 in 1995. Write an equation that will give the price of a math book in any given year, and use this equation to predict the price of the book in 2010.

MORE APPLICATIONS

Solve the following problems.

Exercise 73

Solve for xx size 12{x} {} and yy size 12{y} {}.
y=3x+4y=3x+4 size 12{y=3x+4} {}
y=5x−2y=5x−2 size 12{y=5x - 2} {}

Solution

x=3x=3 size 12{x=3} {}, y=13y=13 size 12{y="13"} {}

Exercise 74

Solve for xx size 12{x} {} and yy size 12{y} {}.
2x−3y=42x−3y=4 size 12{2x - 3y=4} {}
3x−4y=53x−4y=5 size 12{3x - 4y=5} {}

Exercise 75

The supply curve for a product is y=2000x+13000y=2000x+13000 size 12{y="2000"x+"13000"} {}, and the demand curve is y=−1000x+28000y=−1000x+28000 size 12{y= - "1000"x+"28000"} {}, where xx size 12{x} {} represents the price and y the number of items. At what price will the supply equal demand, and how many items will be produced at that price?

Solution

x=5x=5 size 12{x=5} {}, y=23000y=23000 size 12{y="23000"} {}

Exercise 76

The supply curve for a product is y=300x+9000y=300x+9000 size 12{y="300"x+"9000"} {}, and the demand curve is y=−100x+14000y=−100x+14000 size 12{y= - "100"x+"14000"} {}, where xx size 12{x} {} represents the price and yy size 12{y} {} the number of items. At what price will the supply equal demand, and how many items will be produced at that price?

Exercise 77

A demand curve for a commodity is the number of items the consumer will buy at different prices. It has been determined that at a price of $2 a store can sell 2400 of a particular type of toy dolls, and for a price of $8 the store can sell 600 such dolls. If xx size 12{x} {} represents the price of dolls and yy size 12{y} {} the number of items sold, write an equation for the demand curve.

Solution

y=−300x+3000y=−300x+3000 size 12{y= - "300"x+"3000"} {}
(30)

Exercise 78

A supply curve for a commodity is the number of items of the product that can be made available at different prices. A manufacturer of toy dolls can supply 2000 dolls if the dolls are sold for $8 each, but he can supply only 800 dolls if the dolls are sold for $2 each. If xx size 12{x} {} represents the price of dolls and yy size 12{y} {} the number of items, write an equation for the supply curve.

Exercise 79

The equilibrium price is the price where the supply equals the demand. From the demand and supply curves obtained in the previous two problems, find the equilibrium price, and determine the number of items that can be sold at that price.

Solution

x=5.2 y=1440x=5.2 size 12{x=5 "." 2}  {} y=1440 size 12{y="1440"} {}
(31)

Exercise 80

A car rental company offers two plans. Plan I charges $10 a day and 10 cents a mile, while Plan II charges 14 cents a mile, but no flat fee. If you were to drive 300 miles in a day, which plan is better? For what mileage are both rates equal?

Exercise 81

A break-even point is the intersection of the cost function and the revenue function, that is, where the total cost equals revenue. Mrs. Jones Cookies Store's revenue and cost in dollars for xx size 12{x} {} number of cookies is given by R=.80xR=.80x size 12{R= "." "80"x} {} and C=.05x+3000C=.05x+3000 size 12{C= "." "05"x+"3000"} {}. Find the number of cookies that must be sold so that the revenue and cost are the same.

Solution

x=4000x=4000 size 12{x="4000"} {}
(32)

Exercise 82

A company's revenue and cost in dollars are given by R=225xR=225x size 12{R="225"x} {} and C=75x+6000C=75x+6000 size 12{C="75"x+"6000"} {}, where xx size 12{x} {} is the number of items. Find the number of items that must be produced to break-even.

Exercise 83

A firm producing computer diskettes has a fixed costs of $10,725, and variable cost of 20 cents a diskette. Find the break-even point if the diskettes sell for $1.50 each.

Solution

(8250, 12375)

Exercise 84

Whackemhard Sports is planning to introduce a new line of tennis rackets. The fixed costs for the new line are $25,000 and the variable cost of producing each racket is $60. If the racket sells for $80, find the number of rackets that must be sold in order to break even.

CHAPTER REVIEW

Exercise 85

Find an equation of the x-axis.

Solution

y=0y=0 size 12{y=0} {}
(33)

Exercise 86

Find the slope of the line whose equation is 2x+3y=62x+3y=6 size 12{2x+3y=6} {}.

Solution

−2/3−2/3 size 12{ - 2/3} {}
(34)

Exercise 87

Find the slope of the line whose equation is y=−3x+5y=−3x+5 size 12{y= - 3x+5} {}.

Solution

–3

Exercise 88

Find both the x and y intercepts of the line 3x−2y=123x−2y=12 size 12{3x - 2y="12"} {}.

Solution

4, –6

Exercise 89

Find an equation of the line whose slope is 3 and y-intercept 5.

Solution

y=3x+5y=3x+5 size 12{y=3x+5} {}
(35)

Exercise 90

Find an equation of the line whose x-intercept is 2 and y-intercept 3.

Solution

3x+2y=63x+2y=6 size 12{3x+2y=6} {}
(36)

Exercise 91

Find an equation of the line that has slope 3 and passes through the point (2, 15).

Solution

y=3x+9y=3x+9 size 12{y=3x+9} {}
(37)

Exercise 92

Find an equation of the line that has slope - 3 2 - 3 2 and passes through the point (4, 3).

Solution

3x+2y=183x+2y=18 size 12{3x+2y="18"} {}
(38)

Exercise 93

Find an equation of the line that passes through the points (0, 32) and (100, 212).

Solution

y=9/5x+32y=9/5x+32 size 12{y=9/5x+"32"} {}
(39)

Exercise 94

Find an equation of the line that passes through the point (2, 5) and is parallel to the line y=3x+4y=3x+4 size 12{y=3x+4} {}.

Solution

y=3x−1y=3x−1 size 12{y=3x - 1} {}
(40)

Exercise 95

Find the point of intersection of the lines 2x−3y=92x−3y=9 size 12{2x - 3y=9} {} and 3x+4y=53x+4y=5 size 12{3x+4y=5} {}.

Solution

(3, –1)

Exercise 96

Is the point (3, – 2) on the line 5x−2y=115x−2y=11 size 12{5x - 2y="11"} {}?

Solution

No

Exercise 97

Find two points on the line given by the parametric equations, x=2+3tx=2+3t size 12{x=2+3t} {}, y=1−2ty=1−2t size 12{y=1 - 2t} {}.

Solution

(2,1), (5,–1); Answers will vary

Exercise 98

Find two points on the line 2x−6=02x−6=0 size 12{2x - 6=0} {}.

Solution

(3,0), (3, 1); Answers will vary

Exercise 99

Graph the line 2x−3y+6=02x−3y+6=0 size 12{2x - 3y+6=0} {}.

Solution

The line through (–3,0) and (0,2)

Exercise 100

Graph the line y=−2x+3y=−2x+3 size 12{y= - 2x+3} {}.

Solution

The line through (0,3) and (1,1)

Exercise 101

A female college student who is 60 inches tall weighs 100 pounds, and another female student who is 66 inches tall weighs 124 pounds. Assuming the relationship between the female students' weights and heights is linear, write an equation giving the relationship between heights and weights of female students, and use this relationship to predict the weight of a female student who is 70 inches tall.

Solution

y=4x−140;140y=4x−140 size 12{y=4x - "140"} {};140
(41)

Exercise 102

In deep-sea diving, the pressure exerted by water plays a great role in designing underwater equipment. If at a depth of 10 feet there is a pressure of 21 lb/in2, and at a depth of 50 ft there is a pressure of 75 lb/in2, write an equation giving a relationship between depth and pressure. Use this relationship to predict pressure at a depth of 100 ft.

Solution

y=1.35x+15.2;142.5y=1.35x+15.2 size 12{y=1 "." "35"x+"15" "." 2} {};142.5
(42)

Exercise 103

If the variable cost to manufacture an item is $30, and the fixed costs are $2750, write the cost function.

Solution

y=30x+2750y=30x+2750 size 12{y="30"x+"2750"} {}
(43)

Exercise 104

The variable cost to manufacture an item is $10, and it costs $2,500 to produce 100 items. Write the cost function, and use this function to estimate the cost of manufacturing 300 items.

Solution

y=10x+1500;4500y=10x+1500 size 12{y="10"x+"1500"} {};4500
(44)

Exercise 105

It costs $2,700 to manufacture 100 items of a product, and $4,200 to manufacture 200 items. If x represents the number of items, and y the costs, find the cost equation, and use this function to predict the cost of 1,000 items.

Solution

y=15x+1200;16200y=15x+1200 size 12{y="15"x+"1200"} {};16200
(45)

Exercise 106

In 1980, the average house in Palo Alto cost $280,000 and the same house in 1997 costs $450,000. Assuming a linear relationship, write an equation that will give the price of the house in any year, and use this equation to predict the price of a similar house in the year 2010.

Solution

y=10000x+280000;580000y=10000x+280000 size 12{y="10000"x+"280000"} {};580000
(46)

Exercise 107

The population of Argentina in 1987 was 31.5 million and in 1997 it was 42.5 million. Assuming a linear relationship, write an equation that will give the population of Argentina in any year, and use this equation to predict the population of Argentina in the year 2010.

Solution

y=1.1x+31.5;56.8y=1.1x+31.5 size 12{y=1 "." 1x+"31" "." 5} {};56.8
(47)

Exercise 108

In 1955, an average new Chevrolet sold for $2,400, and a similar Chevrolet sold for $15,000 in 1995. Assuming a linear relationship, write an equation that will give the price of an average Chevrolet in any year. Use this equation to predict the price of an average Chevrolet in the year 2010.

Solution

y=315x+2400;19725y=315x+2400 size 12{y="315"x+"2400"} {};19725
(48)

Exercise 109

Two-hundred items are demanded at a price of $5, and 300 items are demanded at a price of $3. If x represents the price, and y the number of items, write the demand function.

Solution

y=−50x+450y=−50x+450 size 12{y= - "50"x+"450"} {}
(49)

Exercise 110

A supply curve for a product is the number of items of the product that can be made available at different prices. A manufacturer of Tickle Me Elmo dolls can supply 2000 dolls if the dolls are sold for $30 each, but he can supply only 800 dolls if the dolls are sold for $10 each. If xx size 12{x} {} represents the price of dolls and yy size 12{x} {} the number of items, write an equation for the supply curve.

Solution

y=60x+200y=60x+200 size 12{y="60"x+"200"} {}
(50)

Exercise 111

Suppose you are trying to decide on a price for your latest creation - a coffee mug that never tips. Through a survey, you have determined that at a price of $2, you can sell 2100 mugs, but at a price of $12 you can only sell 100 mugs. Furthermore, your supplier can supply you 3500 mugs if you charge your customers $12, but only 500 if you charge $2. What price should you charge so that the supply equals demand, and at that price how many coffee mugs will you be able to sell?

Solution

Price=$5.20; # of mugs=1460Price=$5.20 size 12{"Price"=$5 "." "20"} {};# of mugs=1460 size 12{"# of mugs"="1460"} {}
(51)

Exercise 112

A car rental company offers two plans. Plan I charges $12 a day and 12 cents a mile, while Plan II charges $30 a day but no charge for miles. If you were to drive 300 miles in a day, which plan is better? For what mileage are both rates the same?

Solution

150 miles

Exercise 113

The supply curve for a product is y=250x−1,000y=250x−1,000 size 12{y="250"x - 1,"000"} {} and the demand curve for the same product is y=−350x+8,000y=−350x+8,000 size 12{y= - "350"x+8,"000"} {}, where xx size 12{x} {} is the price and yy size 12{y} {} the number of items produced. Find the following.

  1. At a price of $10, how many items will be in demand?
  2. At what price will 4,000 items be supplied?
  3. What is the equilibrium price for this product?
  4. How many items will be manufactured at the equilibrium price?

Solution

  1. 4500;
  2. 20;
  3. 15;
  4. 2750

Exercise 114

The supply curve for a product is y=625x−600y=625x−600 size 12{y="625"x - "600"} {} and the demand curve for the same product is y=−125x+8,400y=−125x+8,400 size 12{y= - "125"x+8,"400"} {}, where xx size 12{x} {} is the price and yy size 12{y} {} the number of items produced. Find the equilibrium price and determine the number of items that will be produced at that price.

Solution

$12; 6900

Exercise 115

Both Jenny and Masur are sales people for Athletic Shoes. Jenny gets paid $8 per hour plus 4% commission on the sales. Masur gets paid $10 per hour plus 8% commission on the sales in excess of $1,000. If they work 8-hour days, for what sales amount would they both earn the same daily amounts?

Solution

1600

Exercise 116

A company's revenue and cost in dollars are given by R=25xR=25x size 12{R="25"x} {} and C=10x+9,000C=10x+9,000 size 12{C="10"x+9,"000"} {}, where xx size 12{x} {} represents the number of items. Find the number of items that must be produced to break-even.

Solution

600

Exercise 117

A firm producing video tapes has fixed costs of $6,800, and a variable cost of 30 cents per tape. If the video tapes sell for $2 each, find the number of tapes that must be produced to break-even.

Solution

4,000

Exercise 118

A firm producing disposable cameras has fixed costs of $8,000, and variable cost of 50 cents a camera. If the cameras sell for $3.50, how many cameras must be produced to break-even?

Solution

2,667

Exercise 119

The Stanley Company is coming up with a new cordless travel shaver just before the Christmas holidays . It hopes to sell 10,000 of these shavers in the month of December alone. The manufacturing variable cost is $3 and the fixed costs $100,000. If the shavers sell for $11 each, how many must be produced to break-even?

Solution

12,500

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