Linear Modeling and Logic Unit

4.2 Linear Modeling

For each scenario:

Scenario 1

You buy a $200 printer and the ink cartridges cost $25 each. Write an equation for total cost.

y = 25x + 200

Cartridges

Total Cost

0

$200

1

$225

5

$325

10

$450

15

$575

Domain:

Domain: \(\begin{equation} x \geq 0 \end{equation} \)

Range:

Range: \(\begin{equation} y \geq 200 \end{equation} \)

This is a blank coordinate plane. The words label and units are included for both the horizontal axis and the vertical axis to help you to remember to label them correctly. The graph is linear and increases from left to right. The x-axis represents the number of computer cartridges and goes from 0 to 17 counting by 1. The y-axis represents the total cost and goes from 0 to 700 counting by 50. The points that are plotted on the line are (0,200), (1,225), (5,325), (10,450), and (15,575).

If your office budgets $30 per month for printer costs, how many ink cartridges can they buy the year they purchase the printer?

25x + 200 = 360

25x = 16

x = 6.4

The office can buy 6 cartridges for the year.

Scenario 2

You get a $50 iTunes gift card for your birthday. Each song costs $1.29. Write an equation for how much is left on your gift card.

y = 50 - 1.29x

Number of Songs

Amount Left on Card

1

$48.71

5

$43.55

10

$37.10

20

$24.20

30

$11.30

Domain:

Domain: \(\begin{equation} 0 \leq x \leq 38 \end{equation} \)

Range:

Range: \(\begin{equation} 0 \leq y \leq 50 \end{equation} \)

This is a blank coordinate plane. The words label and units are included for both the horizontal axis and the vertical axis to help you to remember to label them correctly. A linear graph that decreases from left to right. The x-axis represents the number of songs that are purchased and goes from 0 to 36 counting by 2. The y-axis represents the amount left on the card and goes from 0 to 60 counting by 5. The points that are plotted on the graph are (1,48.71), (5,43.55), (10,37.10), (20,24.20), and (30,11.30).

What is the y-intercept of your graph?

(0,50)

How many single track songs can you buy with your gift card?

50 - 1.29x = 0

-1.29x = -50

x = 38.76

You can buy 38 songs with a $50 gift card.

Scenario 3

It’s summer and Jill is filling her above ground pool. It takes 25 minutes to raise the water level 4 inches. The water level is already 10 inches. What is the rate Jill is filling the pool in inches per minute?

\( \begin{equation} \frac{4 \text{ inches}}{25 \text{ minutes}} = 0.16 \text{ inches per minute} \end{equation} \)

Write an equation for the depth of the pool depending on time.

\( \begin{equation}y = 10 + 0.16x \text{ or } y = 10 + \frac{4}{25} x \end{equation} \)

Time

Depth of Pool

1

10.16

20

13.2

40

16.4

100

26

200

42

Domain:

Domain: \(\begin{equation} 0 \leq x \leq 237.5 \end{equation} \)

Range:

Range: \(\begin{equation} 10 \leq y \leq 48 \end{equation} \)

This is a blank coordinate plane. The words label and units are included for both the horizontal axis and the vertical axis to help you to remember to label them correctly. A linear graph that increases from left to right. The x-axis goes from 0 to 250 counting by 10. The y-axis goes from 0 to 40 counting by 5. The points that are plotted on the graph are (1,10.16), (20,13.2), (40,16.4), (100,26), and (200,42).

If the pool is 4 feet deep, how much longer will it take Jill to fill the pool?

48 = 10 + 0.16x

38 = 0.16x

x = 237.5

It will take 237.5 minutes to fill the pool.

Scenario 4

The gas tank in your car holds 12 gallons. The dashboard statistics tells you that you get 28 miles per gallon. Write an equation for the amount of gasoline left in your tank depending on how many miles you’ve driven.

\( \begin{equation} y = 12 - \frac{1}{28} x \end{equation} \)

Mileage

Amount of Gas in Tank

5

11.8

50

10.2

100

8.4

175

5.75

250

3.1

Domain:

Domain: \(\begin{equation} 0 \leq x\leq 336 \end{equation} \)

Range:

Range: \(\begin{equation} 0 \leq y \leq 12 \end{equation} \)

This is a blank coordinate plane. The words label and units are included for both the horizontal axis and the vertical axis to help you to remember to label them correctly. A linear graph that decreases from left to right. The x-axis represents the mileage and goes from 0 to 350 counting by 50. The y-axis represents the amount left in the gas tank and goes from 0 to 15 counting by 1. The points that are plotted on the graph are (5,11.8), (50,10.2), (100,8.4), (175,5.75), and (250,3.1).

Your trip meter says you’ve gone 290 miles so far on this tank of gas. Can you make it to work and back (25 miles each way) without stopping for gas?

It is a 50 mile round trip.

You have 1.6 gallons left in your tank.

28 * 1.6 = 46 miles

You will not make it to work and back without putting more gas in the tank.