Math 6
Mathematics homework help
Lesson 3.3
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 models are appropriate when the situation has a constant increase/decrease.
· slope is the rate of change.
· the rate of change (slope) has units in context.
· different representations of a linear model can be used interchangeably.
Students will be able to
· label units on variables used in a linear model.
· make a linear model when given data or information in context.
· make a graphical representation of a linear model.
· make a table of values based on a linear relationship.
· identify and interpret the vertical intercept in context.
In this lesson, you will learn about how linear models (linear equations in context) can be useful in examining some situations encountered in real life. A model is a mathematical description of an authentic situation. You can also say that the mathematical description “models” the situation. There are four common representations of a mathematical relationship.
Four Representations
Four Representations of a Relationship
In the last lesson, you explored how BAC could be estimated. If we were considering a 150 pound male who has had 3 drinks, the equation would simplify to
B=−0.015t+0.0835B=-0.015t+0.0835
This is an equation for a model. This equation is useful because it can be used to calculate BAC given a number of hours (for a 150 pound man who’s had 3 drinks). Equations are useful for communicating complex relationships. In writing equations, it is always important to define what the variables represent, including units. For example, in the equation above, B = Blood alcohol content percentage, t = number of hours since the first drink.
Another way that you could have represented this relationship is in a table that shows values of t and B as ordered pairs. An ordered pair is two values that are matched together in a given relationship. You used a table in the last lesson. Tables are helpful for recognizing patterns and general relationships or for giving information about specific values. A table should always have labels for each column. The labels should include units when appropriate.
Time since first drink (hours) | BAC (%) |
0 | 0.0835 |
1 | 0.0685 |
2 | 0.0535 |
3 | 0.0385 |
A graph provides a visual representation of the situation. It helps you see how the variables are related to each other and make predictions about future values or values in between those in your table. The horizontal and vertical axis of the graph should be labeled, including units.
A verbal description explains the relationship in words. In this case, we can do that by looking at how the values in the formula affect the result. Notice that the person’s BAC begins at 0.0835% (at the time he first consumes the 3 drinks), and whenever t increases by 1, the BAC drops by 0.015%. So we could verbally describe the relationship as: The persons BAC starts at 0.0835%, and drops by 0.015% each hour.
Slope
Linear Equations and Slope
The type of equation we looked at for BAC is a linear equation, or a linear model. Recall that the graph of a linear equation is a line. The primary characteristic of a linear equation is that it has a constant rate of change, meaning that each time the input increases by one, the output changes by a fixed amount. In the example above, each time t increased by 1, the BAC dropped by 0.015%. This constant rate of change is also called slope.
Example: If I were walking at a constant speed, my distance would change 6 miles in 2 hours. We could compute the slope or rate of change as 6 miles2 hours=3 miles1 hour6 miles2 hours=3 miles1 hour = 3 miles per hour.
In general, we can compute slope as change of outputchange of inputchange of outputchange of input .
In the graph shown, we can see that the graph is a line, so the equation is linear. We can compute the slope using any two pairs of points by counting how much the input and output change, and divide them. Notice that we get the same slope regardless of which points we use.
If the output increases as the input increases, we consider that a positive change. If the output decreases as the input increases, that is a negative change, and the slope will be negative.
Some people call the calculation of slope “rise over run“, where “rise” refers to the vertical change in output, and “run” refers to the horizontal change of input.
The units on slope will be a rate based on the units of the output and input variables. It will have units of “output units per input units”. For example,
· Input: hours. Output: miles. Slope: miles per hour
· Input: number of cats. Output: pounds of litter. Slope: pounds of litter per cat
Slope-intercept Equation
Slope Intercept Equation of a Line
The slope-intercept form of a line, the most common way you’ll see linear equations written, is
y=mx+by=mx+b
where m is the slope, and b is the vertical intercept (called y intercept when the output variable is y). In the equation, x is the input variable, and y is the output variable.
Notice our BAC equation from earlier, B=−0.015t+0.0835B=-0.015t+0.0835 , fits this form where the slope is -0.015 and the vertical intercept is 0.0835. The input variable is t and the output variable is B.
Sometimes you’ll see the equation written instead in the form y=b+mxy=b+mx .
The slope tells us a rate of change. As we interpreted earlier, in this equation the slope tells us the person’s BAC drops by 0.015% each hour. Or we could say that the rate of change is -0.015% per hour.
The vertical intercept tells us the initial value of the equation – the value of the output when the input is zero. For the BAC equation, the vertical intercept tells us the persons BAC starts at 0.0835%.
#1 Points possible: 5. Total attempts: 5
Write the equation of a line with slope 3 and y-intercept of (0, 7). Use x as the input variable, and y as the output variable.
#2 Points possible: 14. Total attempts: 5
Terry is skiing down a steep hill. Terry’s elevation, EE, in feet after tt seconds is given by E=3300−40tE=3300-40t. The equation tells us that Terry started skiing and his is by
Graphing
Graphing using Slope and Intercept
In earlier lessons, you graphed a formula by calculating points. You certainly can continue to do that, and knowing that an equation is linear just makes that easier by only requiring two points. But it can be helpful to think about how you can use the slope to graph as well.
Suppose we want to graph y=−23x+5y=-23x+5 using the vertical intercept and slope.
The vertical intercept of the function is (0, 5), giving us a point on the graph of the line.
The slope is −23-23 . This tells us that every time the input increases by 3, the output decreases by 2. In graphing, we can use this by first plotting our vertical intercept on the graph, then using the slope to find a second point. From the initial value (0, 5) the slope tells us that if we move to the right 3, we will move down 2, moving us to the point (3, 3). We can continue this again to find a third point at (6, 1). Finally, extend the line to the left and right, containing these points.
#3 Points possible: 5. Total attempts: 5
Sketch a graph of y=−32x+2y=-32x+2 by first placing a point at the y-intercept, then use the slope to find a second point.
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Fundraiser
Problem Situation 1: Fundraiser
A sports team is planning a fundraiser to help pay for equipment. Their plan is to sell team T-shirts to friends and family, and they hope to raise $500. Their local screen printing shop will charge them a $75 setup fee, plus $5 per shirt, and they plan to sell the shirts for $15 each.
#4 Points possible: 12. Total attempts: 5
Explore their profit (the amount they bring in from sales minus costs) by filling in the table below. In all cases (including 0 sold) assume they still paid the $75 setup fee.
Number of shirts made and sold | Profit | |
0 | $ | |
1 | $ | |
2 | $ | |
10 | $ |
#5 Points possible: 5. Total attempts: 5
How much additional profit does the team make for each shirt they sell?
$
#6 Points possible: 5. Total attempts: 5
Create a linear model for their profit in terms of the number of shirts they sell. Use P for the profit, and n for the number of shirts they sell.
#7 Points possible: 5. Total attempts: 5
Create a graph of the linear model.
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#8 Points possible: 5. Total attempts: 5
How much money will they raise if they can sell 30 shirts?
$
#9 Points possible: 5. Total attempts: 5
How many shirts will they have to sell to raise their goal of $500?
shirts
Lattes
Problem Situation 2: Daily Latte
A local coffee shop offers a Coffee Card that you can preload with any amount of money and use like a debit card each day to purchase coffee. At the beginning of the month (when you get your paycheck), you load it with $50. Each day, your short soy latte costs $2.63.
#10 Points possible: 5. Total attempts: 5
Estimate if the Coffee Card will last until the end of the month if you purchase a latte every weekday.
· No, it will run out
· Yes, it will last
#11 Points possible: 5. Total attempts: 5
If you purchase a latte every weekday, create a formula for a linear model for the balance, B, on your coffee card after t weekdays.
#12 Points possible: 5. Total attempts: 5
Use the linear model to calculate if your $50 Coffee Card will last until the end of the month.
The card will run out of money after weekdays. (Answer to the nearest whole day)
The value you found in the last question is called the horizontal intercept (sometimes called the x-intercept). It is the point where the output value is zero and the graph crosses the horizontal axis.
HW 3.3
#1 Points possible: 20. Total attempts: 5
Which of the following representations depict a linear model? Justify your decision based on one of the three characteristics listed above.
a. R=t873.36−2.15R=t873.36-2.15 where R is the % of Americans who are retired in the year t. The graph is shown below. Justification:
b. P = 40 + 2L where P = the perimeter of an area in yards and L = the length of the area in yards. Justification:
c. The graph for braking distance versus speed for the coefficient of friction, f = 0.8, and roadway grade of 5% is shown below. Justification:
d.
Time (min) | Distance (ft) |
0 | 3.5 |
2 | 13.5 |
4 | 23.5 |
6 | 33.5 |
e. Justification:
f. The formula for the area of a circle: A = πr2. Justification:
#2 Points possible: 5. Total attempts: 5
A rental car company charges $45$45 plus 2020 cents per each mile driven. Part1. Which of the following could be used to model the total cost of the rental where mm represents the miles driven.
· C=20m+45C=20m+45
· C=45m+20C=45m+20
· C=0.2m+45C=0.2m+45
· C=2m+45C=2m+45
· C=45m+0.2C=45m+0.2
Part 2. The total cost of driving 300300 miles is; $
#3 Points possible: 12. Total attempts: 5
Indicate if each of the following statements is true or false.
a. The slope of a linear model is a ratio that describes how the outputs of the model increase or decrease as the inputs increase.
b. The slope of a linear model changes depending on the values substituted into the model.
c. The horizontal intercept is represented by an ordered pair in which the first value is 0: (0, __).
d. The vertical intercept is sometimes called a starting or initial value.
#4 Points possible: 15. Total attempts: 5
Sheila wants to lose weight for an upcoming wedding. She currently weighs 186 pounds and her goal is to weigh 140 pounds. After consulting with her doctor, she feels she can safely lose 2 pounds per week. The graph tracks the projected weight loss over time.
a. Write an equation for the weight loss trend. Use W = weight (lb) and t = time (weeks).
b. Use your equation to determine how long will it take Sheila to achieve her desired weight goal. weeks
c. What is the slope (and select the correct units)?
#5 Points possible: 15. Total attempts: 5
Ben has $70 in his savings account. He plans to deposit $40 per week to build his account balance.
a. Complete the following equation to represent the amount of money (A) Ben will have in his account after any number of weeks. Let x represent the number of weeks. A =
b. Which of the following values could be the value of the variable in this context?
· 3
· 0
· 18
· -5
c. Ben wants to use his savings to buy a computer for $750. Use your algebraic expression to determine the number of weeks it will take to buy the computer. weeks
#6 Points possible: 6. Total attempts: 5
Suppose you work for a shuttle service. You see the following spreadsheet that gives information about what your company will charge for rides of different lengths. The formula shown is for cell B2. Complete the statement that tells how the fare is calculated based on the number of miles.
The fare is cents per mile plus a $ flat fee.
#7 Points possible: 20. Total attempts: 5
Doctors use intravenous (IV) drips to deliver fluids and medications to patients. Suppose an IV drip starts with 1,000 ml of saline and dispenses fluid at a rate of 2.5 ml per minute.
a. Write an equation for this situation using F to represent the amount of fluid (saline solution) left in milliliters and t to represent the time in minutes. Hint: If you have trouble writing a model, try making a table first.
b. Is this a linear model? Explain your answer.
c. How long will it take to use all of the saline solution? Give you answer in hours and minutes. hours and minutes
d. Which of the following can be used to describe the point you found in Part (c)?
· Horizontal intercept
· Vertical intercept
· Slope
· Rate of change
#8 Points possible: 15. Total attempts: 5
The Bureau of Labor and Statistics published the employment projections for the Heathcare industry in 2011.
Healthcare employment in 2008 was 14,336,000, and is projected to rise to 17,561,600 by 2018.
a. On average, how many healthcare workers will be added each year during this period? workers/year
b. Which of the following can be used to describe the value you found in Part (a)?
· Horizontal intercept
· Vertical intercept
· Slope
c. Write a linear equation for this relationship. Let W = the number of healthcare workers and t = the number of years after 2008.
#9 Points possible: 5. Total attempts: 5
Give the slope and the y-intercept of the line y=−8x−9y=-8x-9. Make sure the y-intercept is written as a coordinate. Slope = y-intercept =
#10 Points possible: 5. Total attempts: 5
Give the equation of the line with a slope of 2727 and a y-intercept of 22.
#11 Points possible: 12. Total attempts: 5
For the line sketched below
12345-1-2-3-4-5246810-2-4-6-8-10xy
a. What is the value of b, the y-intercept? b =
b. What is the value of m, the slope? m =
c. What is the equation for the line?
#12 Points possible: 8. Total attempts: 5
Sketch a graph of y=12x−2y=12x-2
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