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# Applications Using Linear Models

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Practice Applications Using Linear Models
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Solve Real-World Problems by Using Strategies and a Plan

Have you ever gone on a plane trip? Take a look at this dilemma involving flights, times and friends.

Kevin and Laila sat talking at lunch. Then Carmen came over.

“We were talking about what we did this summer. I went to Yellowstone camping and Kevin did this really great project with a group of kids making a garden. What about you? What did you do with your summer?” Laila asked Carmen as she took a drink of milk.

“I went to see my Grandparents. It was a great time, but I barely made it on the day of my flight,” Carmen said munching a carrot.

“Well, it started out fine. I had a 9 pm flight. I knew that I had to be at the airport 2 hours before the flight and that we live one hour from the airport. I needed $1 \frac{1}{2}$ to pack and take a shower and stuff like that. It would have been fine except I had a plan to play soccer at the park first. I got home at 4:00 and barely made it to the airport,” Carmen explained.

Kevin looked at Laila.

“You should have had plenty of time,” Kevin said.

How does Kevin know this? Can you follow Kevin’s thinking? Why does Kevin make this statement? To figure this out, you will need to apply your problem solving skills.

### Guidance

When problem-solving, you will use strategies as part of a plan. For each situation, you will be asked to read and understand a given problem. You will make a plan to solve by choosing an appropriate strategies. There are multiple ways to solve a word problem. Therefore, you will need to consider and compare different approaches for each problem given.

When you read to understand a problem, you are working to determine what the problem is asking you to do. It helps to highlight the question in the problem. You may want to also underline clue words that may help you with planning and strategy.

A lizard ate five hundred flies on five consecutive nights. Each night he ate twenty-five more than the night before. How many flies did the lizard eat each night?

For this problem, you are to determine the number of flies the lizard ate each night. Here is what you are told:

• You are told that the lizard ate a total of five hundred flies over the course of five nights.
• You are told that the lizard eats twenty-five more flies each night than the night before.
• You should know the word consecutive means a logical sequence or succession. In this case, it means one night after the other.

Make plan to solve the problem.

Use a verbal model to write and solve an equation to determine the unknown number of flies eaten each day.

You are told that the lizard ate a total of five hundred flies in five days. You are also told that each night he eats twenty-five more than the night before. To determine the number of flies consumed each night, you must first determine the number of flies the lizard ate the first night. After determining the number of flies consumed the first night, add twenty-five more each day to get the daily total.

Verbal Model:

number of flies eaten on night one + (number of flies eaten on night one + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five) + (number of flies eaten on night one + twenty-five + twenty-five + twenty-five + twenty-five) = total number of flies eaten over five nights (500)

Let “ $x$ ” represent the number of flies eaten on night one.

The next step is to solve the problem.

Equation:

$x + (x + 25) + (x + 25 + 25) + (x + 25 + 25 + 25) + (x + 25 + 25 + 25 + 25) = 500$

Solution:

$x + (x + 25) + (x + 25 + 25) + (x + 25 + 25 + 25) + (x + 25 + 25 + 25 + 25) &= 500\\x + (x + 25) + (x + 50) + (x + 75) + (x + 100) &= 500\\5x + 250 &= 500$

Next, we solve the equation. Subtract 250 from 500 and divide by 5.

$x=50$

The lizard consumed 50 flies the first night. To determine the number of flies the lizard ate on nights two, three, four, and five, substitute 50 for “ $x$ ” in the equation.

$x + (x + 25) + (x + 50) + (x + 75) + (x + 100) &= 500\\50 + (50 + 25) + (50 + 50) + (50 + 75) + (50 + 100) &= 500\\50 + 75 + 100 + 125 + 150 &= 500$

You can see that on night one, the lizard ate 50 flies.

On night two, the lizard consumed 75 flies.

On night three, the lizard ate 100 flies.

On night four, the lizard ate 125 flies.

On the last night, the lizard ate 150 flies.

Check the Results

You can check your work, by adding the number of flies consumed each night. The sum should be equal to five hundred.

$50 + 75 + 100 + 125 + 150 = 500$

Night One: 50

Night Two: 75

Night Three: 100

Night Four: 125

Night Five: 150

Now look at this situation and answer each question.

A ten year olds’ heart beats approximately 85 times per minute. How many times does the heart beat in 24 seconds?

#### Example A

What is the problem asking you to do?

Solution: Knowing that the heart beats 85 times per minute, determine the number of heart beats in 24 seconds.

#### Example B

What is the plan?

Solution: Since you are being asked to determine an unknown rate, write a proportion to solve.

#### Example C

Can you write an equation and solve this problem?

Solution: Let “ $x$ ” = the number of heart beats in 24 seconds

$& \ \underline{\;\; 85 \ beats\;\;} = \underline{\;\;\;\;\;\;\; x \;\;\;\;\;\;\;}\\& \ 60 \ seconds \quad \ 24 \ seconds\\& \qquad 85(24) = 60(x)\\& \ \quad \underline{\;\; 2,040 \;\;} = \underline{\;\; 60x \;\;}\\& \ \qquad \ 60 \qquad \quad \ 60\\& \qquad \qquad \ x=34 \ beats$

Now let's go back to the dilemma from the beginning of the Concept.

First, why did Kevin make that statement?

Kevin figure out that Carmen should have left for the airport at 6 pm. If she only needed $1 \frac{1}{2}$ hours to get ready, then she should have had plenty of time because she got home at 4 pm leaving her 2 hours to get ready.

Here is the breakdown of her time.

9 pm flight – 2 hours check in = 7 pm

7 pm – 1 hour drive time = 6 pm

$6 - 1 \frac{1}{2}$ hours to get ready = 4:30 pm

Carmen should have left her home at 4:30 pm to be on time for the flight.

### Vocabulary

Ratio
a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.
Proportion
formed when two ratios are equivalent. We compare two ratios, they are equal and so they form a proportion.

### Guided Practice

Here is one for you to try on your own. Notice that some key points are underlined for you.

A train’s caboose is 12 feet long . Each of the train’s eight cars are twice the length of the caboose . Determine the length of the entire train.

Solution

You are to determine the entire length of the train. You were told some information.

• The caboose is 12 feet long.
• There area of an additional eight cars.
• Each car is twice the length of the caboose.

Make a plan to solve the problem.

You can draw a diagram and use a verbal model to visualize the information given in the problem. Then, write an equation to determine the length of the entire train.

Verbal Model:

Eight trains twice the length of the caboose + the length of the caboose = entire length of the entire train

Let “ $x$ ” represent the unknown length of the train.

Equation:

$8(2 \cdot 12) + 12 = x$

Solution:

$8(2 \cdot 12) + 12 &= x\\8(24) + 12 &= x\\192 + 12 &= x\\204 &= x$

The entire train is 204 feet.

### Practice

Directions: Read each problem and then solve each problem.

Ted has a collection of rare coins. He already had 34 coins in his collection. The first week, Ted purchases 1 new coin. During the second week, Ted purchases 4 coins. During the third week, Ted adds 9 new coins to his collection. At this rate how many weeks will it take Ted to collect 125 coins?

1. Which strategy should Ted use to solve this problem?
2. What could Ted draw to help him with his solution?
3. How long will it take Ted to collect 125 coins?

Savannah wants to buy a pair of jeans that cost $59.00. They are on sale for 25% off. 1. Which strategy could Savannah use to calculate the price? 2. What is the amount of the discount? 3. What is the sale price? Carlos was in charge of organizing cookies for a bake sale. He organized them into bundles of six cookies. When he was done, he had 15 bundles of cookies. How many cookies did he start with? 1. Which strategy could you use to solve this problem? 2. Write an equation to describe the problem. 3. Solve the equation. 4. How many cookies did Carlos begin with? Veronica made brownies. She made twice as many brownies as Carlos had cookies. How many brownies did she make? 1. Which strategy could you use to solve this problem? 2. Write an equation to describe the problem. 3. Solve the equation. 4. How many brownies did Veronica make? If Veronica sold half of the cookies that she made, how many would be sold? If she charged$1.50 per brownie, how much money would she make?

1. Which strategy could you use to solve this problem?
2. How much money would she make?