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From 1950 through 2005, the per capita consumption $ C $ of cigarettes by Americans (age 18 and older) can be modeled by $ C = 3565.0 + 60.30t - 1.783t^2, 0 \le t \le 55 $, where $ t $ is the year, with $ t = 0 $ corresponding to 1950.

(a) Use a graphing utility to graph the model.(b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warninghad any effect? Explain.(c) In 2005, the U.S. population (age 18 and over) was 296,329,000. Of those, about 59,858,458 weresmokers. What was the average annual cigarette consumption per smoker in 2005? What was theaverage daily cigarette consumption per smoker?

a) See graphb) $4075, y e s$c) $7366,20$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

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So we have a model for consumption of cigarettes and are some consumption is per capita for per person who's in United States. And we have 3565 plus thesixties 0.3 x and then minus going back looking at my calculator 1.7 83 x squared. And we know that the X value is in time after a time of zero is for 1950 and we know this is supposed to be from zero is less than or equal to X is less than or equal to 55. So this would go. This would equate to 1950 this is going to equate to 2005 and we want to put it on our calculator. And so I looked at I put this in as wise up one in my calculator, and then I did a table of values. I surely knew that my grab had to go up to at least 3565 but more likely my graph was going to curve up, so I didn't know how high to make my grab. So I set my window to go from on X from zero to build that paper on away 0 to 55 and I scaled by fives. And then I went up to 4500 02 4500 and I scaled by five hundreds on this access. So I scaled by 500 this way, and I scaled by fives that way. And when I did, my graph kind of looked something like this. So I get my scatter plot, but use that table to help you find out how high you need to make your window. Then on Part B, we want to look at where that maximum consumption waas. We wanted that point so we could go to second and calculated to a left bound right bound, and we can get that point. And that occurs at 16.9 and 4075 so that maximum consumption is 4075 cigarettes per person at that time. And that would equate to a year of about 1967. And they talked about the idea that they started putting warning labels on cigarettes in 1966. So it looks like we did have a downward trend. So it looks like that advertising was helping that, uh, consumption go down now in part C, We're looking at 2005 and we know that if we look at our graph for our table that the consumption rate waas one for 1487.9 cigarettes per person and we know that our population was nearly 300 million. So if we take that number and multiply it by the population 2000 297 million, 329,000 people that once we get that, that's the total number of cigarettes that are being used. However, we want to take that total and divided by the number of smokers to find out what the rate of consumption is just for smokers. And so we do that division. So again, taking what the consumption rate is times the total population of people at that time. And that actually again tells me how many cigarettes are consumed. But they're consumed by this population of people. These are are smokers. And so when we take this times this and divided by that will end up finding out what the per capita per per smoker rate is and we find out that that is 7366 approximately cigarettes per smoker. That's what the averages. And they said, Well, then what does that equate to per day? Well, this is our yearly consumption. So if we take that 7366 divided by assuming it's not a leaf here dividing by 365 we find out that the average smoker is smoking about 20 cigarettes per day in 2005. Her day and we're all finished. Don't smoke.

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