Linear Modeling and Logic Unit

4.3 Writing Equations of Lines

Introduction

On August 5, 2010, a gold-and-copper mine collapsed in Chile, trapping 33 miners underground. Rescuers used a drilling rig to reach the miners under the earth’s surface. An escape capsule carried each miner up the shaft at a constant rate as shown in the graph.

The Miner Rescue graph is a linear graph that is increasing from left to right. The x-axis represents the time in seconds and goes from 0 to 800 counting by 100. The y-axis represents the depth in feet and goeas from -2500 to 500. Two points on the graph are (0,-2050) and (680,0).

  1. At what underground depth were the miners trapped?

    2. 2040 feet underground

  2. How long has a miner been ascending in the escape capsule when he reaches a depth of 300 feet?

    3. 580 seconds

  3. How long did it take to bring each miner to the surface?

    4. 680 seconds

  4. At what average rate did the escape capsule carry each miner up the shaft?

    5. \( \begin{equation} \frac{-300-0}{580-680}=\frac{-300}{-100}= \text{ 3 feet per second} \end{equation} \)

  5. Write an expression for the depth of the miner during his rescue with respect to time ascending.

    6. y = 3x - 2040

Group Assignment

  1. Use the graph “People With Health Insurance,” which estimates the U.S. population with health insurance in the given years, to answer the following questions.

    The graph is linear and increases from left to right. The x-axis represents the number of years since 1989 and goes from 0 to 6 counting by 1. The y-axis represents the number of poeple in millions and goes from 0 to 250. The points plotted on the graph are (0,212), (1,214), (2,216), (3,218), (4,220), and (5,222).
    1. How many people had health insurance in 1990?

      2. 214 million

    2. How many people had health insurance in 1991?

      3. 216 million

    3. At what rate is the number of people with health insurance increasing each year?

      4. 2 million per year

    4. According to the graph, how many people would you expect to have had health insurance in 1988?

      5. 210 million

    5. According to the graph, how many people would you expect to have had health insurance in 2014?

      6. 262 million

    6. The actual number of people in the U.S. with health insurance in 2014 was about 276 million. Why do you think this differs from your estimate in question (e)?

      7. The function is probably not a linear relationship or the actual number could reflect that the Affordable Care Act was enacted in 2013.

  2. Use the graph “Josh’s Car Trip” to answer the following questions.
    The graph is linear and decreases from left to right. The x-axis represents the time in hours and goeas from 0 to 4 counting by 1. The y-axis represents the number of miles from Josh's home and goes from 0 to 160 counting by 20. Two points on the graph are (0,160) and (4,0).
    1. Is Josh traveling away from home or towards his home during his trip?

      2. toward home

    2. How many miles long is Josh’s trip?

      3. 160 miles

    3. How long does Josh’s trip take?

      4. 4 hours

    4. What is Josh’s speed during his trip?

      5. 40 mph

    5. At this rate, how long would it take Josh to complete a 3300 miles trek across the United States? If he could only drive 8 hours a day, how many days would it take him?

      6. 82.5 hours which is 10.3 days when he is driving 8 hours per day.

  3. Use the graph “Demand Curve” to answer the following questions. The Demand Curve graph is linear and is decreasing from left to right. The x-axis represents the cost in dollars and goes from 0 to 400 counting by 100. The y-axis represents the units sold and goes from 0 to 80,000 counting by 20,000. Two points on the graph are (0,70000) and (350,0)
    1. When the cost of the product is $100, how many units are sold?

      2. 50,000 units

    2. When the cost of the product is $200, how many units are sold?

      3. 30,000 units

    3. When the cost of the product is $300, how many units are sold?

      4. 10,000 units

    4. Is demand increasing or decreasing with respect to cost?

      5. decreasing

    5. What is the rate of change for units sold for every dollar increase?

      6. -200 units per dollar

    6. Which brings in the most money: selling the product for $100, $200, or $300? (Take into account how many units are sold at each price)

      7. Selling the product for $200 brings in the most money.

    7. At what cost does the demand hit zero?

      8. $350

  4. Use the graph “Cost of Cell Phone by Number of Months” to answer the following questions. The cost of the cell phone graph is linear and increases from left to right. The x-axis represents the number of months and goeas from 0 to 13 counting by 1. The y-axis represents the cost in dollars and goes from 0 to 550 counting by 50. Two points on the graph are (6,300) and (11,500).
    1. What two variable quantities are shown on the graph?

      2. number of months and cost of the cell phone

    2. Which is the dependent quantity?

      3. cost of the cell phone

    3. What is the domain as shown on the graph?

      4. \( \begin{equation} 0 \leq x \leq 13 \end{equation} \)

    4. What is the range as shown on the graph?

      5. \(\begin{equation} 0 \leq y \leq 550 \end{equation}\)

    5. What is the initial cost of the cell phone (at month zero)? What might this amount represent?

      6. $60

      This cost may represent taxes or a down payment for the cell phone.

    6. What is the cost of the cell phone at 12 months?

      7. $540

    7. What is the cost of the cell phone at 6 months?

      8. $300

    8. How much does the person pay towards his cell phone per month?

      9. $40

    9. Write an equation for the cost of the cell phone based on number of months?

      10. y = 40x + 60

  5. Use the graph “Cost of Black and Gray Cabs” to answer the following questions. The green cab versus blue cab graph is two increasing linear graphs that intersect at one point. The x-axis represents the number of miles and goes from 0 to 14 counting by 1. The y-axis represents the total cost in dollars for each of the cabs and goes from 0 to 14 counting by 1. Two points on the line for the green cab are (0,2) and (5,11). Two points on the line for the blue cab are (0,5) and (10,14). The point of intersection of the two lines in (3.333,8).
    1. What two variable quantities are shown on the graph?

      2. number of miles and total cost

    2. What are the units for the dependent quantity?

      3. dollars

    3. If your destination is 10 miles away, which cab would you call?

      4. The black cab

    4. If your destination is 2 miles away, which cab would you call?

      5. The gray cab

    5. At what mileage does cost not matter when choosing a cab?

      6. 3.3 miles

    6. How much does the gray cab charge per mile?

      7. $1.80

    7. Write an equation for the cost of the gray cab.

      8. y = 1.80x + 2

    8. How much does the black cab charge per mile?

      9. $0.90

    9. Write an equation for the cost of the black cab.

      10. y = 0.90x + 5

  6. Use the graph “Length of Spring by Weight Attached” to answer the following questions. The Length of Spring by Weight Attached graph is linear and increases from left to right. The x-axis represents the weight in pounds and goes from 0 to 6 counting by 1. The y-axis represents the length in inches and goes from 0 to 30 counting by 5. Two points on the graph are (2,14) and (5,23).
    1. What is the domain of the length of spring by weight attached?

      2. \(\begin{equation} 0 \leq x \leq 7 \end{equation}\)

    2. What is the range of the length of spring by weight attached?

      3. \(\begin{equation} 0 \leq y \leq 30 \end{equation}\)

    3. What is the initial length of the spring?

      4. 8 inches

    4. How much more does the spring stretch with 5 pounds of weight compared to 2 pounds of weigh?

      5. 9 inches more

    5. How much does the spring stretch per additional pound of weight added?

      6. 3 inches per pound

    6. Write an equation for the total length of the spring with respect to weight attached.

      7. y = 3x + 8

    7. Predict the length of the spring when 8 pounds are attached to it.

      8. 32 inches

  7. Use the graph “Money Earned” to answer the following questions. The Money Earned graph is linear and increases from left to right. The x-axis represents the years played and goes from 0 to 5.5 counting by 0.5. The y-axis represents the money earned in millions of dollars and goes from 0 to 100 counting by 20. The points that are plotted on the graph are (1,10), (2,30), (3,50), (4,70) and (5,70).
    1. How much does the player earn per year?

      2. $20 million

    2. If the linear trend were extended, how much would the player earn at 0 years? Does this make sense?

      3. According to the graph, the player would earn -$10 million, which does not make sense.

    3. Write an equation for the total money earned by the player based on years played.

      4. y = 20x - 10 (in millions)