gallons (g) Find a formula which calculates the price (P) from the number of gallons pumped (g).

Transcription

1 Homework about Linear Models: (1) On the way home, you stop for gas. The price per gallon is $3.39. If we view the number of gallons you pumped as "input g", g = 0,1,2,3,4,... and if the other quantity is the "price", what would be the pattern created? gallons (g) Price ($) Find a formula which calculates the price (P) from the number of gallons pumped (g). (2) On Sundays, Charles sells the Cincinnati Enquirer down on Victory Parkway, at the Gilbert intersection. The Sunday Paper costs $1.50 a piece. Charles takes $20 in quarters for change, and puts them into a bucket next to the stacks of newspapers. Every time he sells a paper, he puts $1.50 into the bucket (and keeps the tip if the customer pays more than $1.50). (a) We want to capture the relationship between the number of papers sold, and the amount of money in the bucket using a table. Produce such a table. (b) We want to capture the relationship between the number of papers sold, and the amount of money in the bucket using a formula connecting "n", the number of papers sold and "M", the amount of money in the bucket. Find such a formula.

2 (3) The yearly number of Chapter 12 banruptcy filings in the US for the years from 1996 to 2000 can be well imitated by the formula B = t, where B is the number of bankruptcy filings and t is the number of years past (So, for the year 1996: t = 0, for the year 1997: t = 1, etc.) (a) Show the relationship between t and B in a table. (b) What does the " 1063" in the formula tell us about US bankruptcy filings? (You should answer this in a sentence or two.) (c) What does the fact that the formula contains the " t "-part tell us about the number of bankruptcy filings in the US during the time period ? (You should answer this in a sentence or two.)

3 First off, remeber from now on, that models (patterns, formulas,functions) are referred to as " LINEAR" if the model (function/pattern/formula) " adds or subtracts the same amount to the "output" (the calculated value) each time the "input" (step#, year, whatever) goes up by 1. A few comments/reminders: (optional reading) We have talked in class about WHY graphed pairs of the form (input, output) for any such linear rule or pattern will all lie on a single straight line. If we add the same amount over and over, the graph will go "up" (from left to right), if we subtract the same amount over and over, the graph will go "down" from left to right. If we add MORE each step to the right (increase of input by 1) then that straight line we see when we graph (input, output)-pairs will get steeper! How much we add or subtract is often called "slope" of the line. It is also referred to as the "rate of growth". For example: To imitate Xavier tuition data, we used the calculator to find the linear pattern Xœ&!$*Þ#$ $("Þ($ >, where we interpret > as the number of years after 1992, and X is the tuition in US $. This is a LINEAR function (model, pattern...), because for the year 1992 (which is when >œ! ), the pattern starts with value of &!$*Þ#$ (when > œ!), and then the model adds $("Þ($ each time the input ( > ) increases by one. The calculated value ("output") X is interpreted as tution, and that value goes up by $("Þ($ each time the input ( > ) goes up by 1, which is interpreted as "one year later". In other words: Our model acts by "increasing the tuition by $ each year". People will often say that "The tuition is going up at a rate of dollars per year", which explains the fact that this "number which is added each step" is often called "rate of growth". It indicates how fast the tuition (output) is growing per input unit (year, in this case.) If we graph the calculated tuition values, we start with at >œ!, then we move one > -unit to the right and the next point will be units higher than the first point. Then we move another > -unit to the right, and the third point will be exactly $("Þ($ units higher than the previous (second) point, and so on and on... This is why the points (input-output-pairs) will all lie on a straight line when graphed. If we add more each step right, the line gets steeper. (etc.) This is why/how that number by which the input is multiplied influences the steepness of the straight line, and the word "slope" is used to indicate the steepness. Keep these ideas and connections between reality - formula - graph - number pattern in mind when you work on the following problems, all of which are taken from the text. Also: The text uses different language from the language we have used in class so far. This is why you have to read slowly and thoughtfully, and with my comments above in mind you will have to try to make sense of and get used to these words. (Problems start on the next page)

4 (4) On a certain tomato farm, the quantity of tomatoes harvested in any given year is given by Xœ' & V, where Vis the amount of rainfall during the growing season in inches, and X is the total tomato harvest measured in hundreds of pounds. (a) Interpret the numbers which appear in this formula (' and &) in the context of tomato harvest and rainfall. You should write at least one sentence for each of the numbers. What does the "'" in the formula tell you about the relationship between rainfall and Tomato Harvest?? What does the "&" in the formula tell you about the relationship between rainfall and Tomato Harvest?? (b) Graph a few (R, T) - pairs. Do they lie on a straight line? Which of the two numbers in the formula (& or 'Ñ influences how steep the line is? How does this "influence" work? (5) Under overcrowding conditions at an assembly line for BMW-sunroofs, productivity of the assembly line will is WÐBÑ œ "(!! #!B, where " B" is the number of of workers above the number of workers needed to operate the assembly line, and " W" is the number of sunroofs produced in aday. (a) In a few, clear sentences, explain what the formula says about the assembly line's productivity. Be as exact as possible. Avoid vague statements that could be made less vague. (b) Graph a few (input, output)-pairs, that is ( Bß W)-pairs. Why is the straight line on which these points lie go downhill from left to right? Explain why that happens. How much does the line drop when you increase the input ( B) by 1? What does it mean (in the context) "to increase the input ( B) by 1"? (c) What number is called the "slope of the line" which emerges when you graph the model? At what rate is the output changing per input unit? (Write a sentence in response, and fill in what input and output are, instead of saying "input" and "output". In the next two exercises, use the information given to find a function (formula) which expresses the connection between the two quantities of interest. Decide which of the two should be viewed as output. (The other one is then input.) (6) The population of some country was "(& million in 2004, and has since increased by 150,000 people each year. (7) Fabric sheeting ismanufactured on a loom at a rate of square feet per minute. The first %Þ(& six square feet of the fabric is unusable. (We care about "usable fabric" only.)

5 (8) The picture below shows the graph of a linear function (lots and lots of input-output-pairs) used to model yearly US industrial carbon dioxide emissions, in million metric ytons, between 2004 and (a) Estimate the slope of the graph (by checking how much the line moves up when you move one input unit (to the right). Report this answer, but don't talk about the slope of the line. Talk about carbon dioxide emissions... (b) Are emissions increasing, decreasing, or constant? (Stupid question, right?) Explain how this is reflected in the value of the slope in (a). What would be different in the graph, if the emissions were decreasing? What would the graph look like if the emissions had been constant? (9) The number of people who use the internet in the US > years after 2008 is projected to follow the model MÐ>Ñ œ &Þ'% > "*$Þ%% million users. (a) At what rate is the number of Internet users in the US growing, according to this model? (b) Evaluate the formula for the input >œ!. In a sentence, explain what this calculated output means in the context. (c) Draw a picture similar to the one in the previous exercise, and use the same method (go one step to the right...) to determine the slope. (d) You were probably taught that "slope" is "rise/run" for any two points on the line. Pick the points for >œ" and >œ% on the line. Calculate rise/run. Is your result the same as what you got in (c)? If it is not, why not? If it is, why is that?

6 (10) In 1960, the birth rate for US women aged was 59.9 births per thousand women. In 2006, the birth rate for US women aged was 41.9 births per thousand women. (a) Calculate the rate at which the "number of births per thousand US women aged 15-19" had changed during the indicated time period, assuming that this rate was constant. (You need to figure out what this means! Doing it is easy...) Try to report this rate in a clean and proper English sentence (or two.) This is not easy. (b) Write a model, a formula, for the birth rate for US women aged for this time period. Don't forget to indicate what the input and output "mean" and how they are measured. (c) Use the model from (b) to estimate/predict the birth rate for US women aged in the year (Is this called an "extrapolation" or an "interpolation"? Look up what these words mean! Whichever one this is, what would be an example for the other? In all of the remaining Problems, use the LinReg procedure on your calculator to find a Linear Model for the given data. Do not forget to ALWAYS explain what the letters/variables in your model mean (e.g. T is tution in US$, t is the # years since 1992, etc.) (11) The pet industry is big business in the US. The amount spent yearly on pets is staggering, and seems to be ever increasing. Here is a table of values. (The left column shows "Year", and the right column shows Expenditures (in billion dollars)): (a) Make a scatterplot of the data in your calculator. Based on the scatterplot: Why is a linear model appropriate? (b) Align the data so that 1994 corresponds to " >œ! ". Use your LinReg procedure to find a linear model for the amount of money spent each year on Americans on pets. (c) Use the model to estimate the amount of money that will be spent on American pets in the year What assumption must be made for this to be considered a valid extimate?

7 (12) The table below shows the number of people in Europe who use as part of their jobs: (left column shows the calendar year. The right column shows # of users in millions) (a) Use the LinReg procedure to find a model for the number of European business users. When you report this model, what else is needed besides the formula? Provide that as well! (b) Use sentences to explain what this model says about how the number of European users has developed between 2005 ad This verball information should describe how the model imitates the data. How does the MODEL see the # of European business users develop? Is that actually true? Explain. (c) Use your model from (a) to estimate the number of European business users in Is this estimate called an "interpolation" or an "extrapolation"? Why?