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| [[jobs]] | | [[jobs]] |
| [[math2]] | | [[math2]] |
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| In this chapter, you’ll learn how to
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| solve problems involving more than
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| one equation, like finding out when
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| two animals that grow at different
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| rates will be the same size.
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| The costs for parking in two different parking garages are given
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| in the table at the right.
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| Two hikers are walking along a marked trail. The first hiker starts at a
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| point 6 mi from the beginning of the trail and walks at a speed of 4 mi>h. At the
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| same time, the second hiker starts 1 mi from the beginning and walks at a speed
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| of 3 mi>h.
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| The number of right-handed students in a mathematics class
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| is nine times the number of left-handed students. The total number of students
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| in the class is 30. How many right-handed students are in the class? How many
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| left-handed students are in the class?
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| A plant nursery is growing a tree that is 3 ft tall and grows at an average rate
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| of 1 ft per year. Another tree at the nursery is 4 ft tall and grows at an average rate of
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| 0.5 ft per year. After how many years will the trees be the same height?
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| At a local fitness center, members pay a $20 membership fee and $3 for
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| each aerobics class. Nonmembers pay $5 for each aerobics class. For what number
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| of aerobics classes will the cost for members and nonmembers be the same?
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| Tickets for a concert cost $10 each
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| if you order them online, but you must pay a service
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| charge of $8 per order. The tickets are $12 each if you
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| buy them at the door on the night of the concert.
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| Scientists studied the weights of two alligators over a period of 12 months.
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| The initial weight and growth rate of each alligator are shown below. After how many
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| months did the alligators weigh the same amount?
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| One satellite radio service charges $10 per month plus
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| an activation fee of $20. A second service charges $11 per
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| month plus an activation fee of $15. In what month was
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| the cost of the service the same?
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| Two professional downhill skiers are racing
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| at the speeds shown in the diagram. Skier 1
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| starts 5 s before Skier 2. The course is
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| 5000 ft long. Will Skier 2 pass Skier 1?
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| How do you know?
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| You are looking for an after-school job. One job pays $9 per
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| hour. Another pays $12 per hour, but you must buy a uniform that costs $39. After
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| how many hours of work would your net earnings from either job be the same?
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|
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| A cell phone provider offers a plan that costs $40 per month plus
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| $.20 per text message sent or received. A comparable plan costs $60 per month but
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| offers unlimited text messaging.
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| A school is planning a field trip for 142 people. The trip will use six
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| drivers and two types of vehicles: buses and vans. A bus can seat 51 passengers. A
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| van can seat 10 passengers. Write and solve a system of equations to find how many
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| buses and how many vans will be needed.
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| The measure of one acute angle in a right triangle is four times the
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| measure of the other acute angle. Write and solve a system of equations to find the
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| measures of the acute angles.
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| Adult tickets to a play cost $22. Tickets for children cost $15.
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| Tickets for a group of 11 people cost a total of $228. Write and solve a system of
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| equations to find how many children and how many adults were in the group.
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| A farmer grows corn, tomatoes, and sunflowers on a 320-acre farm.
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| This year, the farmer wants to plant twice as many acres of tomatoes as acres of
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| sunflowers. The farmer also wants to plant 40 more acres of corn than of tomatoes.
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| How many acres of each crop should the farmer plant?
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| Michelle and Pam are running a 200-m race. Michelle runs at an
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| average of 7.5 m/s. Pam averages 7.8 m/s, but she starts 1 s after Michelle.
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| The rectangle shown has a perimeter of 34 cm and the given area.
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| Its length is 5 more than twice its width. Write and solve a system of equations
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| to find the dimensions of the rectangle.
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| A toy store worker packed two boxes of identical dolls and plush toys
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| for shipping in boxes that weigh 1 oz when empty. One box held 3 dolls and 4 plush
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| toys. The worker marked the weight as 12 oz. The other box held 2 dolls and 3 plush
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| toys. The worker marked the weight as 10 oz. Explain why the worker must have
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| made a mistake.
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| A hotel offers two activity packages. One costs $192 and includes 3 h of
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| horseback riding and 2 h of parasailing. The second costs $213 and includes 2 h of
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| horseback riding and 3 h of parasailing. What is the cost for 1 h of each activity?
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| Each of the squares in the figures shown at the right has the
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| same area, and each of the triangles has the same area. The total area of
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| Figure A is 141 cm2. The total area of Figure B is 192 cm2. What is the area
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| of each square and each triangle?
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| The zoo has two other water tanks that are leaking. One tank contains
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| 10 gal of water and is leaking at a constant rate of 2 gal/h. The second tank
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| contains 6 gal of water and is leaking at a constant rate of 4 gal/h. When will
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| the tanks have the same amount of water? Explain.
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|
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| break even pt
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| A fashion designer makes and sells hats. The material for each hat costs
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| $5.50. The hats sell for $12.50 each. The designer spends $1400 on advertising. How
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| many hats must the designer sell to break even?
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| A puzzle expert wrote a new sudoku puzzle book. His initial costs are $864.
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| Binding and packaging each book costs $.80. The price of the book is $2.
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| How many copies must be sold to break even?
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|
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| identifying constraints
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| The local zoo is filling two water tanks for the elephant exhibit. One water tank
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| contains 50 gal of water and is filled at a constant rate of 10 gal/h. The second water
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| tank contains 29 gal of water and is filled at a constant rate of 3 gal/h. When will the
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| two tanks have the same amount of water? Explain.
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| You row upstream at a speed of 2 mi>h. You travel the same distance
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| downstream at a speed of 5 mi>h. What would be your rowing speed in
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| still water? What is the speed of the current?
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| b. Reasoning Suppose your rowing speed in still water is 3 mi>h and
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| the speed of the current is 4 mi>h. What happens when you try to
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| row upstream?
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| Suppose you graph a system of linear equations and the intersection
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| point appears to be (3, 7). Can you be sure that the ordered pair (3, 7) is the
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| solution? What must you do to be sure?
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| A group of scientists studied the effect of a chemical on various strains
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| of bacteria. Strain A started with 6000 cells and decreased at a constant rate of
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| 2000 cells per hour after the chemical was applied. Strain B started with 2000 cells
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| and decreased at a constant rate of 1000 cells per hour after the chemical was
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| applied. When will the strains have the same number of cells? Explain
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| A traveler is walking on a moving walkway in an airport. The
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| traveler must walk back on the walkway to get a bag he forgot. The traveler’s
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| groundspeed is 2 ft>s against the walkway and 6 ft>s with the walkway. What is
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| the traveler’s speed off the walkway? What is the speed of the moving walkway?
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| A kayaker paddles upstream from camp to photograph a waterfall and
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| returns. The kayaker’s speed while traveling upstream and downstream is shown
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| below. What is the kayaker’s speed in still water? What is the speed of the current?
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| You have a jar of pennies and quarters. You want to choose 15 coins that
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| are worth exactly $4.35.
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| a. Write and solve a system of equations that models the situation.
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| b. Is your solution reasonable in terms of the original problem? Explain.
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| You can represent the value of any two-digit number with the
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| expression 10a 1 b, where a is the tens’ place digit and b is the ones’ place digit.
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| For example, if a is 5 and b is 7, then the value of the number is 10(5) 1 7, or 57.
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| What two-digit number is described below?
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| • The ones’ place digit is one more than twice the tens’ place digit.
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| • The value of the number is two more than five times the ones’ place digit.
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| You want to sell 1-lb jars of mixed peanuts and cashews for $5. You
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| pay $3 per pound for peanuts and $6 per pound for cashews. You plan to combine
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| 4 parts peanuts and 1 part cashews to make your mix. You have spent $70 on
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| materials to get started. How many jars must you sell to break even?
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| Last year, one fourth of the students in your class played an instrument. This year,
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| 6 students joined the class. Four of the new students play an instrument. Now, one
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| third of the students play an instrument. How many students are in your class now?
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|
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| inequalities
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| You are planning what to do after school. You can spend at most
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| 6 h daily playing basketball and doing homework. You want to spend less than 2 h
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| playing basketball. You must spend at least 1
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| 12
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| h on homework. What is a graph
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| showing how you can spend your time?
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|
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| Suppose you have a job mowing lawns that pays $12 per hour. You
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| also have a job at a clothing store that pays $10 per hour. You need to earn at least
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| $350 per week, but you can work no more than 35 h per week. You must work a
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| minimum of 10 h per week at the clothing store. What is a graph showing how
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| many hours per week you can work at each job?
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|
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| Two friends agree to split the driving on a road trip from Philadelphia,
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| Pennsylvania, to Denver, Colorado. One friend drives at an average speed of
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| 60 mi>h. The other friend drives at an average speed of 55 mi>h. They want to drive
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| at least 500 mi per day. They plan to spend no more than 10 h driving each day. The
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| friend who drives slower wants to drive fewer hours. What is a graph showing how
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| they can split the driving each day?
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|
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| You are fencing in a rectangular area for a garden. You have
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| only 150 ft of fence. You want the length of the garden to be at least 40 ft. You want
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| the width of the garden to be at least 5 ft. What is a graph showing the possible
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| dimensions your garden could have?
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| • What variables will you use? What will they represent?
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| • How many inequalities do you need to write?
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|
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| A jeweler plans to produce a ring made of silver and gold. The price of
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| gold is about $25 per gram. The price of silver is approximately $.40 per gram. She
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| considers the following in deciding how much gold and silver to use in the ring.
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| • The total mass must be more than 10 g but less than 20 g.
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| • The ring must contain at least 2 g of gold.
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| • The total cost of the gold and silver must be less than $90.
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| a. Write and graph the inequalities that describe this situation.
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| b. For one solution, find the mass of the ring and the cost of the gold and silver.
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| A teacher wants to post a row of student artwork on a wall that is 20 ft
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| long. Some pieces are 8.5 in. wide. Other pieces are 11 in. wide. She is going to leave
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| 3 in. of space to the left of each art piece. She wants to post at least 16 pieces of art.
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| Write and graph a system of inequalities that describes how many pieces of each
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| size she can post.
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