End of Chapter Big Problems

End of chapter “Big Problems”

Chapter 1: Modeling Chicago’s Population

Chicago Population [From official decennial census figures] Year Population

1830 100 1840 4,470 1850 29,963 1860 112,172 1870 298,977 1880 503,185 1890 1,099,850 1900 1,698,575 1910 2,185,283 1920 2,701,705 1930 3,376,438 1940 3,396,808 1950 3,620,962 1960 3,550,404 1970 3,369,357 1980 3,005,072 1990 2,783,726 2000 2,896,016 2010 2,695,598

This graph of Chicago’s population was made from the data above. Use all that you have learned in these lessons to describe and analyze this graph fully and correctly.

Is this a graph of a function? Explain why or why not.

Is it a one-to-one function? Explain why or why not.

What is the domain?

What is the range?

Over which interval(s) is it increasing?

Over which interval(s) is it decreasing?

Is there a relative maximum? Relative minimum?

Identify any x or y intercepts. Is this an even function? Or an odd function? Or neither?

Choose an interval that will have a positive rate of change and calculate it. Explain the meaning of that rate of change.

Choose an interval that will have a negative rate of change and calculate it. Explain the meaning of that rate of change.

Choose an interval that will have a rate of change equal or close to zero. Explain the meaning of that rate of change.

Write the equation of the line that would pass through the first data point and the last data point.

Would that linear model be appropriate to use to predict the population of Chicago in the future? Why or why not?

If viewing this graph on your calculator, what would be an appropriate window?

What quadrant is this in?

Extra Credit Assignment—Research and write a report on the changes seen in Chicago’s population over time, specifically the drop in population since 1950. (Earn up to four points on Test 1, based on thoroughness, accuracy, and quality of writing. Cite all sources used.)

Chapter 1: Modeling Temperature

The following function is the formula used to convert temperature measured in Fahrenheit, f, to Celsius, C: .

The hottest day in Chicago in the last 80 years was recorded on July 13, 1995 as part of the heat wave the killed over 700 people. The temperature reached 106°F that day. The coldest day in Chicago was on January 20, 1985. The temperature was -27°F. Convert these temperatures to Celsius.

Write the inverse function of the formula.

Confirm that the equations are inverses of each other.

Explain the meaning and use of the inverse equation.

What is the slope of this line (inverse function) and what is the meaning of the slope in this context? Charles' Law states that the volume of a gas has a linear relationship with the temperature of the gas assuming a constant mass and a constant pressure. As the temperature increases, the volume increases.

A 600 mL sample of Nitrogen is at 27⁰ Celsius. When the sample is heated to 77⁰ Celsius, the volume increases to 700 mL. Write the linear function that expresses volume, V, as a function of the temperature, C, in Celsius.

Explain in words the transformation of the common linear equation of y = x to your equation from above.

Graph the relationship between volume and temperature.

This equation uses temperature in Celsius. What if the temperature was given in Fahrenheit? Write the equation that expresses the volume as a function of the temperature in Fahrenheit. Think of this as a composite function V(C(f)).

Extra credit assignment—Research and report on the heat wave in Chicago 1995 and the people who died as a result of the heat. Specifically comment on the inequity in the deaths in geographical, racial, and socioeconomic terms. (Earn up to four points on Test 1, based on thoroughness, accuracy, and quality of writing. Cite all sources used.)

Chapter 2: Modeling the Chicago Homicide Rate

Take a minute to examine this graph.

Describe how Chicago’s homicide rate has changed over time. (Include in your description the maximum and minimum values and the intervals on which the graph is increasing or decreasing.) Comment on the data: What did you learn about homicide in Chicago by reading this graph? Does anything surprise you? How is this graph different from the graph of the number of homicides that we examined a few weeks ago?

What could be added to the graph to better understand and analyze this data?

What type of model is this? What do the variables in the equation represent?

What is the overall trend? What is the rate of change (slope) of the model’s equation? What is the meaning of this rate of change?

Calculate the rate of change from 1870 to 2000 using the given data. Does this match the slope of the model?

According to the model, what would have been the homicide rate in 2010? Does it match the data? Explain any differences.

According to the model, what will the homicide rate be in 2020? How accurate do you think this prediction is?

Do you think a linear model is the best model for this data? Why or why not? For what range of years would a linear model be appropriate? What type of equation would be appropriate for the years 1960 to 2010?

What type of model is this?

According to this model, what will the homicide rate be in 2010? Do you think this is a more accurate prediction than the one made with the linear model?

According to this model, when will the homicide rate be five homicides per 100,000 people?

Why do you get two answers? What do they mean in this context? Can you check the accuracy of one of these predictions?

Can the changes in Chicago’s homicide rate be explained using only mathematics? What information would you need to help explain the changes in the rate?

Extra Credit Assignment—Research Chicago’s homicide rate. Report on explanations of the changes in the next class. Bring evidence of your claims and site your sources. (Earn up to four points on Test 1 based on thoroughness, accuracy, and quality of writing.)

Chapter 3: Blood Alcohol Content

About 3,800 people are arrested for driving under the influence of drugs or alcohol in Chicago each year.

BAC (Blood Alcohol Content) is a percentage of ethanol in the blood. A person’s BAC is a function of the amount of alcohol consumed, the person’s gender, the person’s weight, and the time since alcohol was consumed. The legal limit in most states to be considered driving under the influence (DUI) is .08% for drivers over the age of 21 (for those under 21, there is a zero tolerance law—it is a DUI if any alcohol is present in the blood).

Do you think that the BAC is directly or inversely proportional to the amount of alcohol consumed? What about the relationship between BAC and weight?

The BAC varies directly with amount of alcohol consumed and inversely with weight. Write the general equation of variation.

If a 120-pound woman drinks two tequila shots (1.5 oz each of 40% alcohol), her BAC would be . 078%, just barely under the legal limit to drive. First, calculate the amount of alcohol consumed in those two shots. Then, calculate the value of the constant of variation, k. Rewrite the general equation with this constant.

Use the equation to calculate the BAC for a 140-pound woman who drinks four glasses of wine (5 oz per glass of 12% alcohol). Would she be considered legally drunk?

This formula does not yet take into account time. What would happen to the BAC over time? How do you think we need to account for the time since drinking?

The body eliminates alcohol from the system linearly at a rate of about .015% per hour. Rewrite the formula from above to account for this fact.

If it has been two hours since the woman had the four glasses of wine, what would her BAC be then?

How does weight affect the BAC? Assume a man drinks six beers (12 oz each of 5% alcohol) over three hours. Write the equation for BAC as a function of weight. (The constant of variation for men is 7.04.)

Graph this function. Make sure to choose an appropriate window for the graph.

At what weight would this situation lead to driving over the legal limit and risking a DUI?

How many beers can you consume at your weight over two hours to be under the legal limit?

Extra Credit Assignment: Research and report on drunk driving rates in the US. Describe how they have changed over time. Site the source(s) of your data. (Earn up to three points on Test 2 based on thoroughness, accuracy, and quality of writing.) Chapter 4: Chicago Skyway: High Finance

The Chicago Skyway is a toll road that features a gigantic bridge passing over the Kankakee River and an industrial area of south Chicago and Northwest Indiana. The Skyway provides a shortcut from the Dan Ryan Expressway in Chicago to the Indiana Tollway. The skyway opened in 1958 and was run by the Chicago Department of Streets and Sanitation until the year 2005 when it was leased for 99 years for $1.8 billion to Skyway Concessions Company (SCC) owned by a consortium of French investors, Cintra Concesiones de Infraestructuras de Transporte S.A. and Macquarie Atlas Roads and Macquarie Infrastructure Partners. Now, in 2015, that lease is being sold to the Canadian Pension fund for $2.8 billion. The owner of the lease gets to collect the tolls. But, is it worth it? Do these companies really make that much money? How much do they make?

Photo from Cintra Concessiones website.

According to the original agreement, "Tolls for passenger autos on the Skyway are limited under the agreement to no more than $2.50 until 2008, $3.00 until 2011, $3.50 until 2013, $4.00 until 2015, $4.50 until 2017, and $5.00 starting in 2017, unless inflation is higher during that period, with later adjustments equal to the greater of 2% per year, the increase in inflation, or the per capita GDP increase. Limits on commercial vehicles are comparable to passenger autos except that the agreement includes a congestion pricing provision, which permits a further 40% increase in daytime commercial tolls if the operator has a program in place for granting a reduction equal to that amount for commercial vehicles between the hours of 8 p.m. and 4 a.m."

The Skyway currently charges a $4.50 toll for a standard passenger car. Assuming the Skyway will charge a $5.00 toll for a passenger car in 2017, and assuming that tolls will increase 2% per year thereafter, answer the following questions:

a) What will the toll for a standard passenger car be in the year 2020?

b) In what year would the toll be $10?

c) What will the toll be in the year 2097?

It is estimated that the Skyway took in $87 million in revenue in 2014 charging only $4 per car. Consider this model for the revenue function, , where t is the number of years after 2014.

a) What is the growth rate in revenue? According to the model, is this growth compounded yearly or continuously?

b) According to this model, what would the revenue be predicted for the year 2024?

c) According to this model, what would the revenue be predicted for the year 2097?

d) Find the doubling time for this model.

e) In what year would the revenue be four times what it was in 2014?

f) In what year will annual revenue reach $200 million?

g) In what year would it reach $1 billion?

h) It is estimated that the Skyway made $8 million in profit in 2014.

a. For an investment of $1.8 billion, what percent is represented by that $8 million?

b. What were the costs to run the Skyway in 2014?

c. Assuming that costs to run the Skyway increase at a rate of 1% per year, how much profit would be made in 2097?

When the city sold the Skyway in 2005, it socked away $750 million from the sale into a contingency fund. Suppose the fund pays 2.55% interest compounded continuously, how much would be in that fund in 2097?

Extra Credit Assignment—Research Chicago’s history of privatizing public assets such as parking meters and parking garages. Are these good things for the city and the people that live in the city or visit the city? Bring evidence of your claims and site your sources. (Earn up to four points on Test 1 based on thoroughness, accuracy, and quality of writing.)