```Question 462524
In 1960, the average American disposed of 2.7 pounds of garbage per day, whereas in 2003 this amount was 4.3 per day? How do I make this a linear function?====================================
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Linear equations have a starting point and a constant rate of change. Often we see these equation in slope-intercept form.
{{{y=mx+b}}}
where m is b is the y-intercept (starting point), and m is the slope (rate of change.)
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rate of change
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Let's calculate the rate of change for this situation. Since the pounds of garbage per day has changed over time, we can use this for our rate of change. (We're going to assume that the amount Americans throw away has been increasing at a constant rate.).
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We see that an average American threw away 2.7 pounds a day in 1960, and 4.3 pounds a day in 2003. We need to decide if the units of our rate of change will be pounds per day or pounds per year since we have data for 2 points 43 years apart.
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The amount of garbage an average American threw away in a year is probably more useful. Let's figure that out for out two data points.
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We multiply 2.7 x 365 = 985.5 since there are about 365 days in a year. We do the same for 4.3 x 365 = 1569.5. In 1960, an average American disposed of 985.5 pounds of garbage, while in 2003, the average had increased to 1,569.5 pounds per year.
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Now we're ready to calculate the rate of change.
rate of change = [change in amount of garbage thrown away] / [change in years]
m = (1569.5 - 985.5) / (2003 - 1960)
m = 584 / 43
m = 13.581
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Every year, an average American threw away about 13.581 pounds more garbage than the year before.
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Starting point
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The linear equation also has a starting point. For example, in this problem, we use the amount of garbage thrown away per year in 1960: 985.5 pounds
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Writing the Equation
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We'll use the y = mx + b format. Our variables are
x = the number of years after 1960 (i.e. for 1960, x = 0)
y = the pounds of garbage disposed of by an Average American in year x.
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y = mx + b
y = (13.581)x + (985.5)
y = 13.581x + 985.5
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Let's check our equation using our equation with our two data points
In 1960, x = 0
y = 13.581(0) + 985.5
y = 985.5 (true!)
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In 2003, y = 43
y = 13.581(43) + 985.5
y = 1569.5 (true!)
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So a linear equation modelling this situation is y = 13.581x + 985.5
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hope this helps!
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Ms.Figgy
math.in.the.vortex```