Math 7
Mathematics homework help
Lesson 3.5
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).
· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.
· 25. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand that
· linear equations can approximate nearly linear data.
Students will be able to
· find the equation of a line that estimates nearly linear data by calculating the rate of change and the vertical intercept of the line.
· use approximate linear models to interpolate and extrapolate.
Equation of Line Between Two Points
In an earlier lesson, you learned that the equation of line can be written in the form y=mx+by=mx+b . This makes it easy to write the equation when we know the slope and initial value.
In the last lesson, you learned how to compute the slope when you had two points that lie on the line, using the formula slope=change of outputchange of input=y2−y1x2−x1slope=change of outputchange of input=y2-y1x2-x1 .
Now suppose had two points, neither of which was the y intercept, and we want to write an equation for the line. To do that, we need to use the slope and a point to find the intercept.
Example: Find the equation of a line passing through (2,5) and (4, 6).
The slope is change of outputchange of input=y2−y1x2−x1=6−54−2=12change of outputchange of input=y2-y1x2-x1=6-54-2=12
We know the general equation of a line is , so putting in the known slope gives
y=12x+by=12x+b
Now, to find b we can replace x and y with one of our points, and solve for the value of b that will make the equation true. Using (2,5):
5=12(2)+b5=12(2)+b
5=1+b5=1+b
4=b4=b
We have found the y-intercept and can now write the equation for the line: y=12x+4y=12x+4
The process for finding an equation of a line is:
1. Identify two points that lie on the line
2. Calculate the slope between those points. m=y2−y1x2−x1m=y2-y1x2-x1
3. Start with y=mx+by=mx+b . Put in the slope, substitute one of the points for x and y, and solve for b.
4. Write the final equation of the line.
For more about finding the equation of line, here are some videos:
· Finding the equation of a linear in slope-intercept form. [+]
· Another example of finding the equation of a line given two points. [+]
#1 Points possible: 5. Total attempts: 5
Find an equation of line through the points (15,8) and (25,6). Use y for the output, and x for the input.
#2 Points possible: 5. Total attempts: 5
Recall in the last lesson that Raj weighed 193.4 pounds after 0.5 weeks, and 187.6 pounds after 4 weeks. Find a linear model for Raj’s weight, w, after t weeks. Round the slope and intercept to two decimal places if needed.
Minimum Wage
Problem Situation 1: Minimum Wage
The minimum wage is the lowest amount a company is allowed to pay its workers. In the United States, the minimum wage was introduced in 1938. The federal government sets a minimum wage, currently $7.25 per hour, and states and cities can set a higher minimum wage if desired. In Washington State in 2015, the minimum wage was $9.47 per hour, and is adjusted each year based on cost of living.
As you may have read in the newspapers, there are groups working to raise the minimum wage to $12 or $15. Seattle has adopted a plan to increase the minimum wage to $15 over the next several years (by 2017 or 2018 for large employers, and by 2019 or 2021 for smaller companies). How much of a jump is this over the normally expected increases?
To explore this question, we will look at the historical increases of minimum wage in Washington. Real data rarely fall on a straight line, but sometimes data show a definite trend. If the trend is close to linear, the data can be approximated by a linear model. This means that a linear model gives good estimates of what the data will be if the trend continues. A model can also be used to estimate values between data points.In this lesson, you will learn to create linear models from data.
The following data shows the minimum wage in Washington from 2005 to 2015.
Year | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 |
Min Wage | $7.35 | $7.63 | $7.93 | $8.07 | $8.55 | $8.55 | $8.67 | $9.04 | $9.19 | $9.32 | $9.47 |
When data appears approximately linear, we might want to find a linear model to approximate the data.
#3 Points possible: 4. Total attempts: 5
Each of the lines below was found by using a pair of values from the data. Which of the lines is a reasonable approximation of the data (there may be more than one answer)?
·
·
·
·
#4 Points possible: 5. Total attempts: 5
Find an equation for the linear model approximating the minimum wage data, using the data values from 2005 and 2015, letting W be the minimum wage and t the year, measured in years after 2000. Round values to three decimal places if needed.
#5 Points possible: 15. Total attempts: 5
Fill in the blanks below to explain the meaning of the slope and intercept of your equation. The equation tells us that approximately and it has been by
#6 Points possible: 5. Total attempts: 5
Using your model, predict the minimum wage in 2020.
$
#7 Points possible: 8. Total attempts: 5
When does your model predict the minimum wage would reach $15 if it continued increasing at the same rate?
Algebraically, it would happen when t = (give this answer to one decimal place)
Practically, this means in the year (give this to a whole year)
Height Chart
Problem Situation 2: Height Chart
The chart to the below is a growth chart for boys, aged 2 to 15 years. The different curves show different percentiles for growth; the top curve shows the 95th percentile, where 95% of boys are that height or shorter. The middle curve shows the 50th percentile (the median height), and the bottom curve shows the 5th percentile.
#8 Points possible: 5. Total attempts: 5
Find an equation for an approximate linear model for the median height curve, using the data values from ages 3 and 15. Write the height, H, in terms of the age, t.
#9 Points possible: 10. Total attempts: 5
Use your model to predict the height of a 10 year old boy.
cm
How well does your model’s prediction agree with the data in the chart?
· The value is not very close
· The value is very close
#10 Points possible: 10. Total attempts: 5
Use your model to predict the height of a 16 year old boy.
cm
Are you more or less confident in this prediction than the prior one?
· More confident
· Same confidence
· Less confident
#11 Points possible: 10. Total attempts: 5
Use your model to predict the height of a 30 year old man.
cm
How accurate is this prediction likely to be?
· Very accurate
· Somewhat accurate
· Not accurate
#12 Points possible: 8. Total attempts: 5
Determine a reasonable domain for this model. That is, determine an interval of ages for which this model is likely to be reasonably accurate.
≤ age ≤
#13 Points possible: 8. Total attempts: 5
Ninety percent of children fall between the 5th percentile and the 95th percentile. We’ll call this the “typical” interval. Using the chart, write an inequality for the “typical” interval of heights for a 9 year old boy.
cm ≤ height ≤ cm
#14 Points possible: 8. Total attempts: 5
If a lost male child is found who is 145 cm tall, determine an interval of likely ages for the child. Give whole numbers.
≤ age ≤
Summary – Finding Linear Formulas
In the last several lessons you have explored linear equations in a variety of ways. It can feel like there is a lot of different things to remember, but they’re really all related – each lesson we’ve been adding one new layer to what was done in the previous lesson. Here is a summary to tie together everything you’ve learned in the last several lessons.
Linear Equations
Linear equations have the form y=mx+by=mx+b , where
· m is the rate of change, or slope.
· The units will be “outputs per input”. For example if the input is years and outputs is people, the slope will have units “people per year”
· b is the initial value, or vertical-intercept
· The units will be the same as the output quantity. For example if the input is years and outputs is people, the vertical intercept will have units “people”.
To find a linear equation from given information:
· Is the slope (rate of change) given to you?
· If yes, identify it
· If no, calculate it from the given data, using m=change of outputchange of input=y2−y1x2−x1m=change of outputchange of input=y2-y1x2-x1
· Is the vertical intercept given to you?
· If yes, identify it
· If no, plug any given (x,y) point and the slope into the equation y=mx+by=mx+b and solve for b.
· Now that you have the slope and intercept, write the equation.
For examples of all the cases, see this video [+]
HW 3.5
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· It is necessary to be given the starting value of a linear model or it is impossible to find the equation
· It is necessary to have three or more points to find a linear model
· Printing costs can be modeled with a linear equation
· Given the slope and a point, you can solve for the vertical intercept of a linear model
The flight times and distances from Memphis, Tennessee, to various cities are given in the table and graph below. Use this information to answer Questions 2–4.
Time to Fly from Memphis to Various Cities | ||
City | Distance (Mi) | Time (minutes) |
Atlanta | 332 | 82 |
Boston | 1139 | 191 |
Detroit | 611 | 121 |
Baltimore | 787 | 162 |
Cincinnati | 403 | 100 |
Flight times retrieved for flights leaving July 28, 2011 from Delta.com on July 22, 2011.
#2 Points possible: 5. Total attempts: 5
Use what you know about the relationship of the equation to the graph to select the model from the list below that is most representative of the data. Do not actually find a linear model to make your choice.
· t = -0.13d + 110 with d representing the distance in miles and t representing the time in minutes.
· t = 0.13d + 40 with d representing the distance in miles and t representing the time in minutes.
· t = -0.13d + 40 with d representing the distance in miles and t representing the time in minutes.
· t = 0.13d + 110 with d representing the distance in miles and t representing the time in minutes.
#3 Points possible: 6. Total attempts: 5
Using the model you chose in the previous problem, estimate the flight time from Memphis to Philadelphia if the distance between the two cities is 874 miles. Round to the nearest minute.
hours and minutes
#4 Points possible: 5. Total attempts: 5
Based on the model you chose, select the response that best tells how long it will take a Delta plane once in the air flying at full speed to travel 1 mile. Hint: The model includes both flying time and time on the ground.
· It takes 0.13 minutes or around 8 seconds to go one mile.
· It takes 40.13 minutes to go one mile.
· The plane goes 0.13 miles in one minute so it takes between 7 and 8 minutes to go one mile.
· It takes 40 minutes to go one mile.
The table below shows the world record in the men’s 100-meter dash from 1912 to 2009. Use this to answer Questions 5–8. (Note: A change to electronic timing in the 1970s might explain the 30-year gap in records from 1968 to 1999.)
Men’s 100=meter dash 2 | |
Year | World Record Time (nearest tenth) |
1912 | 10.6 |
1921 | 10.4 |
1930 | 10.3 |
1936 | 10.2 |
1956 | 10.1 |
1960 | 10 |
1968 | 9.9 |
1999 | 9.8 |
2008 | 9.7 |
2009 | 9.6 |
#5 Points possible: 5. Total attempts: 5
Create a scatter plot on paper – it’s best to do this on graph paper – using the data relating year and world record time on the 100 meter dash. Let the horizontal axis represent years, t, where t is the number of years after 1900. The vertical axis represents R, the time in seconds of the record 100 meter dash. Use a straight edge to draw the line you think is the best linear model for this data. Write the equation of that line here.
R =
#6 Points possible: 5. Total attempts: 5
Use the model you created to predict when the world record will fall below 9.0 seconds.
The model predicts the world record will fall below 9.0 seconds in year
#7 Points possible: 12. Total attempts: 5
Identify whether the following statements are true or false based on your model. Note that the values given are approximations.
a. On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a minute each year.
b. On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a second each year.
c. On average, the men’s world record in the 100-meter dash has decreased about one-tenth of a second every 10 years.
#8 Points possible: 8. Total attempts: 5
Models often only work well for a limited range of input values. Outside that range, the model is said to break down. Which of these explains why your world record model might break down? There may be more than one correct answer.
· Eventually the model will predict negative times for the race.
· The times are not really accurate.
· Changes in equipment or training might make for sudden improvements in times and change the trend in the current data.
· The data do not form a perfect line.
#9 Points possible: 5. Total attempts: 5
The following chart gives sunrise times for New York City in hours after midnight as measured in Eastern Standard Time (EST) on the 15th of each month. The first month is August 2010, the second month is September 2010, and so on. (Note: Since you are looking at general trends, scales are not included on this graph.)
Would it be appropriate to use a linear model to represent these data? Explain.
In baseball, a team must score more runs than its opponent to win the game. To score runs players must reach base. This is measured by the on-base percentage. A player who reaches a base may or may not score. The following graph compares a team’s on-base percentage with the runs scored. The scatterplot gives data relating the number of runs and the on-base percentage. Use the graph to answer Questions 10 and 11.
#10 Points possible: 5. Total attempts: 5
Katy used the graph to create a linear model that she could use to predict the number of runs scored using the on base percentage. Her model was r = 30b + 450 where, r = number of runs and b = on-base percentage (e.g., for 32%, b=32). What is wrong with Katy’s model?
· 450 runs is not the vertical intercept because the scale on the horizontal axis does not start at 0.
· Katy put the rate and the vertical intercepts in the wrong places in the equation.
· The data are not nearly linear so a linear model should not be used.
#11 Points possible: 5. Total attempts: 5
Based on the slope in Katy’s model, r = 30b + 450, if on-base percentage for a team goes up by 1% then by about how much does the number of runs scored go up?
· 450 runs
· 0.03 runs
· 30 runs
#12 Points possible: 20. Total attempts: 5
The graph below shows the cost of basic cable service from a cable provider from 2002 to 2010.3
a. A representative of the company made a linear equation of the data using the 2002 price as a starting value and the 2005 price as a second data point. What is the company’s equation? Use Pc for the price per month and t for time in years since 2002. Round the slope to the nearest hundredth.
b. A consumer advocate made a linear equation of the data using the 2002 price as a starting value and the 2009 price as a second data point. What is the advocate’s equation? Use Pa for the price per month and t for time in years since 2002. Round the slope to the nearest hundredth.
c. What will the price per month be in 2015 based on the company’s model? The consumer advocate’s model? Hint: Remember that t is the number of years since 2002. Company’s projection: $ Consumer advocate’s projection: $
d. Why might the company and the consumer advocate choose those particular points to make their models?
e. Which of the following is the most accurate estimate of the relative increase from 2002 to 2010?
· 90-100%
· 75-85%
· 50-60%
· 35-45%
#13 Points possible: 5. Total attempts: 5
Find the equation (in terms of xx) of the line through the points (-2,-9) and (5,5) yy =
#14 Points possible: 5. Total attempts: 5
Find the equation (in terms of xx) of the line through the points (-4,3) and (1,5)
y=y=
#15 Points possible: 5. Total attempts: 5
A company has a manufacturing plant that is producing quality canisters. They find that in order to produce 130 canisters in a month, it will cost $3840. Also, to produce 370 canisters in a month, it will cost $8160. Find an equation in the form y=mx+b,y=mx+b, where xx is the number of canisters produced in a month and yy is the monthly cost to do so. Answer: y=y=