Plan Systems Models

Lesson Plan: Modeling real world (and mathematical) situations with systems of equations – 2009

1. Objectives Specific objectives: model situations involving

students will

(a) Converging and diverging travellers

• Define variables x number of hours since first start, y distance travelled. y = 20x A’s locationy = 30(x − 2) B’s location

(b) Attendance + income (c) Limited supplies (d) Place value and number theory with systems of equations and inequalities 2. Agenda • Guided exploration of six problems • Student practice of six problems • Reflection • Assignment hw with solutions 3. Six Problems to review together. Converging Rates: Converging travellers A drives towards B at 35 miles per hour. B drives towards A at 50 miles per. B leaves 20 minutes before A and they are 80 miles apart. How long until they cross paths? How many miles will A and B have driven? • Model: Draw two points, label A and B, label the distance (80 miles) and the starting times as 0 hours and 31 of an hour. • Define variables. x represents the amount of time since B started driving, y represents the drivers’ distance from A’s starting location, A starts at 0, B starts at 80.
A’s location (y) can be modeled by 35 x − 13 , Choose x − 31 because A loses 20 minutes of driving time by starting later. B’s location should then be modeled 80 − 50x. To find out when (x) and where (y) they cross paths solve the system of equations   1 y = 35 x − 3 y = 80 − 50x • Answer They will cross after approximately 55 1 hour and 4.71 minutes, 51 ≈ 1.0784 hours, and .0784hr = 4.704min. A will have traveled 35(1.0784 − .33333) ≈ 26.1 miles, and B will have travelled 50 · 1.0784 = 53.9 miles. Playing catch up A drives from X towards Y at 20 miles per hour, B drives from X towards Y at 30 miles per hour. If A leaves two hours before B, and X and Y are three hundred miles apart, when will X and Y cross paths? • Draw X and Y, draw two arrows from X towards Y, label one A and one B, apply rate information

• Answer: 6 hours after A left they’ll cross paths. Each will have travelled 120 miles. Diverging Travellers A drives 45 miles per hour, B walks 3 miles per hour in the opposite direction, and they start at the same time, how long will it take for A and B to be 100 miles apart? • Draw a point and two lines extending in opposite directions, label the lines A and B, apply rate information • Define variables x number of hours since they began, notice that this can be solved using a simple equation 45x + 3x = 100 • A system of equations might look something like y = 3x y = −45x In order to answer the question from this system you’d have to find the x-value at which the y coordinates are 100 apart. This is not especially simple or effective. • Answer After exactly 2 hours and 5 minutes. 1 5 = 2 60 100 ÷ 48 = 2.083 = 2 12 Attendance and income Tickets to Game A cost $15, and for game B they are $25. 777 tickets were bought all together, and $14,925 was made. How many tickets were bought to each game? • Draw two tickets, list info • Define variables: A, and B, the number of tickets sold to each game A + B = 777 15A + 25B = 14925 • Answer: Game A: 450 tix, 327 to game B. Limited Supplies There are 105 gallons of liquid A and 207 gallons of liquid B. Mix1 contains 5 gallons of A and 11 gallons of B and Mix2 contains 7 gallons of A and 10 gallons of B. How many units of Mix1 and Mix2 can be made? • List the supplies, sketch out each mixture and what it contains.

• Define variables x number of Mix1 ’s and y number of Mix2 ’s. Determine that an inequality would best represent the solution set, because as long as you are using no more than the total supplies, there are many possible variations of the amount of Mix1 and Mix2 . Setup one inequality for each limiting factor, 5x + 7y counts the number of gallons of Liquid A in Mix1 and Mix2 . 5x + 7y ≤ 105

Liquid A

4x + 10y ≤ 207

Liquid B

• Answer: Because there are many solutions, solutions are best represented graphically, see below.

Place Value and Number Theory When the digits of a two digit number are reversed the resulting number is 27 less than the original. The sum of the digits is 13, what is the original number? • Diagram the problem. x,y then y,x. Pick two numbers 1 and 3, how do you represent the value using only the digits? 10(1) + 1(3) or 10(1) + 3. • Define variables: x original tens digit, y original ones digit. 10x + y = 10y + x + 27 x + y = 13 • Answer: x = 8, y = 5 so 85 is the original number.

15 10 5 5

10

15

Six Problems 1. Converging travellers A drives towards B at 35 miles per hour. B drives towards A at 50 miles per. B leaves 20 minutes before A and they are 80 miles apart. How long until they cross paths? How many miles will A and B have driven? • Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions 2. Playing catch up. A drives from X towards Y at 20 miles per hour, B drives from X towards Y at 30 miles per hour. If A leaves two hours before B, and X and Y are three hundred miles apart, when will X and Y cross paths? • Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions 3. Diverging Travellers A drives 45 miles per hour, B walks 3 miles per hour in the opposite direction, and they start at the same time, how long will it take for A and B to be 100 miles apart? • Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions 4. Attendance and income Tickets to Game A cost $15, and for game B they are $25. 777 tickets were bought all together, and $14,925 was made. How many tickets were bought to each game?

• Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions 5. Limited Supplies There are 105 gallons of liquid A and 207 gallons of liquid B. Mix1 contains 5 gallons of A and 4 gallons of B and Mix2 contains 7 gallons of A and 10 gallons of B. How many units of Mix1 and Mix2 can be made? • Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions 6. Place Value and Number Theory When the digits of a two digit number are reversed the resulting number is 27 less than the original. What is the original number? • Draw a diagram to represent the problem

• Write a system of equations to model the information

• Solve the system

• Answer the questions

HW Model each problem algebraically (as an equation, system of equations, or system of inequalities, answer each question 1. Tortoise and Hare The Tortoise travels at 25 feet per minute, while the hare can do 25 feet in 5 seconds. The Tortoise gets a head start of 10 minutes. After the head start, about how many seconds will it take the Hare to catch the Tortoise?

2. Capital Crossing From Dover, Dellaware to Frankfort, Kentucky (both state capitals) it is approximately 669 miles by car. One car leaves Frankfort for Dover at 66 miles per hour, and another leaves Dover for Frankfort and travels at 64 miles per hour, approximately how far from Dover will they cross paths? How long will it take them to cross paths?

3. Create distance Two friends begin at the same spot and ride in opposite directions. Friend1 rides at 15 miles per hour and friend two at 17 miles per hour. How long until they are 8 miles apart?

4. School Lunch A school lunch offers students two options. Option #1: two pieces of pizza and 1 piece of fruit. Option #2: One piece of pizza and two pieces of fruit. The school buys 50 pizzas (each with 8 slices) and 12 boxes of fruit (each with 25 pieces) each day. How many option #1’s and #2’s can the school offer daily?

5. Digits The digits in a two digit number sum to 12, when they are reveresed the outcome is 18 less than the original number. What was the original number?

6. Concert Tickets for Michael Jackson’s concerts in London’s O2 Arena beginning on July 8th of 2008 ranged from approximately $200 to about $320 dollars. Jackson expected to make about $50 million dollars from his 50 shows. The O2 arena has a capacity of 20,000 seats. If twenty percent of the revenue from each show went to him, and his expectations were correct, about how many people paying $200 and $320 did he need in attendance each night?

HW Solutions Model each problem algebraically (as an equation, system of equations, or system of inequalities, answer each question 1. Tortoise and Hare The Tortoise travels at 25 feet per minute, while the hare can do 25 feet in 5 seconds. The Tortoise gets a head start of 10 minutes. After the head start, about how many seconds will it take the Hare to catch the Tortoise? x number of minutes from Tortoise start time, y distance travelled. Note, convert the Hare’s speed into feet per minute to match units with the Tortoise, so the Hare travels 300 feet per minute. y = 25x

Tortoise

y = 300(x − 10)

Hare

25x = 300x − 3000 −275x = −3000 x = 10.90 It will take about 11 minutes 2. Capital Crossing From Dover, Dellaware to Frankfort, Kentucky (both state capitals) it is approximately 669 miles by car. One car leaves Frankfort for Dover at 66 miles per hour, and another leaves Dover for Frankfort and travels at 64 miles per hour, approximately how far from Dover will they cross paths? How long will it take them to cross paths? x number of hours of travelling for each car, y distance from Dover y = 669 − 66x

Car from Fankfort

y = 64x 64x = 669 − 66x

Car from Dover

130x = 669 19 x=5 130   669 ≈ 329.3538462 y = 64 130 19 So ≈ 5.15 hours, (exactly 5 130 hours). The cars will be about 329.35 miles from Dover.

3. Create distance Two friends begin at the same spot and ride in opposite directions. Friend1 rides at 15 miles per hour and friend two at 17 miles per hour. How long until they are 8 miles apart? x number of hours of riding. 15x + 17x = 8 32x = 8 1 8 = x= 32 4 After a quarter hour (15 minutes) Friend1 and Friend2 will be 8 miles apart. 4. School Lunch A school lunch offers students two options. Option #1: two pieces of pizza and 1 piece of fruit. Option #2: One piece of pizza and two pieces of fruit. The school buys 50 pizzas (each with 8 slices) and 12 boxes of fruit (each with 25 pieces) each day. How many option #1’s and #2’s can the school offer daily? Let x represent the number of Option #1’s and y, the number of Option #2’s.

180 160 140 120 100 80 2x + y ≤ 50 · 8 x + 2y ≤ 12 · 25

Limited by Pizza Limited by Fruit

60

Pizza

20

y ≤ 400 − 2x 1 y ≤ 150 − x 2

40

Fruit

20 40 60 80 100 120 140 160 180

5. Digits The digits in a two digit number sum to 12, when they are reveresed the outcome is 18 less than the original number. What was the original number? x is tens place, y ones place.

x + y = 12 10x + y = 10y + x + 18 x = 12 − y 10(12 − y) + y = 10y + (12 − y) + 18 120 − 9y = 9y + 30 18y = 90 y=5 x=7 So the original number was 75. 6. Concert Tickets for Michael Jackson’s concerts in London’s O2 Arena beginning on July 8th of 2008 ranged from approximately $200 to about $320 dollars. Jackson expected to make about $50 million dollars from his 50 shows. The O2 arena has a capacity of 20,000 seats. If twenty percent of the revenue from each show went to him, and his expectations were correct, about how many people paying $200 and $320 did he need in attendance each night? Let x represent the number of people paying $200 each night, and y, the number paying $320.

50(200x + 320y) ≥ 50, 000, 000 · 5 10, 000x + 16, 000y ≥ 250, 000, 000 10x + 16y ≥ 250, 000 x + y ≤ 20000

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