Cab companies often charge a flat fee for picking someone up and then charge an additional fee per mile driven. Pick a U.S. city and research the rates of two different cab companies in that city. Find companies that charge different amounts per mile and have different flat fees. Task 1 a. For the first company, express in words the amount the cab company charges per ride and per mile. b. Write an equation in slope-intercept, point-slope, or standard form. Explain why you chose the form you did. c. What do the x-intercept and y-intercept mean in the context of this problem? Hint: What do you pay when you step into the cab? Task 2 For the second company, express in a table the cost of the cab ride given the number of miles provided. Number of Miles 0 1 2 3 4 5 Total Cost (dollars) a. Write an equation in slope-intercept, point-slope, or standard form. Explain why you chose the form you did. b. What does the slope mean in the context of the problem? Task 3 Cabs use a valuable commodity—gas! Research average gas prices from 2005– 2015 for the city you chose. a. Create a table showing the average gas price each year. b. Create a scatter plot of the data in your table. c. What equation models the data? What are the domain and range of the equation? Explain how you determined your answers. d. Is there a trend in the data? Does there seem to be a positive correlation, a negative correlation, or neither? How much do you expect gas to cost in 2020? Explain.
For Task 1, let's choose New York City as the U.S. city and research the rates of two different cab companies.
First Company:
a. The first company charges a flat fee of $3.50 for picking someone up and an additional fee of $2.50 per mile driven.
b. Let's write the equation in slope-intercept form. We can represent the total cost (y) in dollars as a function of the number of miles driven (x):
y = 2.50x + 3.50
I chose slope-intercept form because it is the most common and straightforward way to represent a linear equation.
c. In the context of this problem, the x-intercept (where y = 0) represents the number of miles that would result in a $0 cost. This is not possible in this case since there is always a flat fee of $3.50. The y-intercept (where x = 0) represents the cost when no miles are driven, which is the flat fee of $3.50.
For Task 2, let's consider the second cab company:
Second Company:
Number of Miles: 0 1 2 3 4 5
Total Cost (dollars): $4.00 $6.00 $8.00 $10.00 $12.00 $14.00
a. We can observe that the cost increases by $2.00 for every mile driven. Let's write the equation in slope-intercept form:
y = 2.00x + 4.00
I chose slope-intercept form again because it is the most suitable form for a linear equation.
b. In the context of this problem, the slope of 2.00 means that for every additional mile driven, the cost increases by $2.00.
For Task 3, let's now research average gas prices for New York City from 2005-2015:
a. Average Gas Prices in New York City (Dollars per Gallon):
Year Average Gas Price (dollars)
2005 2.41
2006 2.98
2007 3.14
2008 3.88
2009 2.59
2010 2.85
2011 3.76
2012 4.00
2013 3.93
2014 3.44
2015 2.78
b. Let's create a scatter plot of the data.
c. The best-fitting equation for the data seems to be a line that decreases from 2008 to 2015. Let's use the linear regression equation to model the data:
y = -0.0477x + 99.706
In this equation, x represents the year from 2005 to 2015, and y represents the average gas price in dollars per gallon. The domain is the set of years {2005, 2006, ..., 2015}, and the range is the set of average gas prices.
d. There is a negative correlation in the data. From 2005 to 2008, the average gas price increased, and from 2008 to 2015, it gradually decreased.
In 2020, we can make a prediction based on the trend in the data. Assuming the decreasing trend continues, we can expect gas prices to be lower than the highest prices recorded between 2008 and 2015.