from-1950-through-2005-the-per-capita-consumption-c-of-cigarettes-by-americans-age-18-and-older-can-

Question

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 warning
had any effect? Explain.
(c) In 2005, the U.S. population (age 18 and over) was 296,329,000. Of those, about 59,858,458 were
smokers. What was the average annual cigarette consumption per smoker in 2005? What was the
average daily cigarette consumption per smoker?

Sheryl Ezze · Accepted Answer

So we have a model for consumption of cigarettes and are some consumption is per capita for per person who&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;re all finished. Don&#x27;t smoke.

Ankit Gupta · Answer

So in this question we have, the CD is equal to minus 1.3 90 square plus 36.5 p plus 3 +871 where he is from 5 to 60. Onda City denotes and well for capita cigarette consumption. On this, the the North&#x27;s the time in years after 1990 as given in the question. So using a graphing calculator driven a graph, this figure I&#x27;ll be having something like this. So this is the X axis which denotes T, and this is the why access, which is going to denote SETI that is, per capita annual cigarette consumption. So when I draw the graph in this given interval 5 to 60 I will have it something like this. The end values are a living told me the nature of the graph will be something like this. Okay, now in the second, but using the same grab. If I have to approximate the here when the maximum and we&#x27;ll consumption is happening, then I&#x27;ll find ordered is equal to 13.13 What maximum see the value? A lot of late I&#x27;d see the value, So that means if he&#x27;s called 0 13.13 That means the heiress here when maximum consumption is happening. All said that here is 2000 today because, as given in the question, it is eight years ahead of 1990 also from the same grab mike and interpret corresponding to this year the maximum every general consumption will be happening. So maximum and well consumption is a quarto. I&#x27;ll says See off 13.13 This one new happens to be coming out to be 4110 So even the graph ill in bigger the same thing. 13.13 cover for 110 All right now in the hard part that asking Beginning in 1996 all cigarette packets were acquired by Lord to carry out health warnings, booting the warning at any effect. Actually, starting from this period represents 1955 right in the 1966 Basically, somewhere over here, 1966 will be somewhere warrior. So clearly, I can see that in spite of having warning inspired off health warnings, the cigarette consumption, consumption, murder capital kept increasing and will be as it can be observed from the graph but it can be observed here also that later, after 2003 the warnings had some effect as the and we&#x27;ll consumption but covered up started decreasing. So we are interpreting all this information from the graph that we have obtain only know in the next question they&#x27;re asking me and, well, harder, average and, well, cigarette conduction. In that year, 2010 will be given by Sea Off 20 because from 1990 it is 20 years ahead. So either substitute Easy called 20 in the given expression that I heard this miners 1.39 p square plus 36.5 p plus 3 +871 So on substituting this I&#x27;ll be having sea or 20 is equal to 4050 All right now, this is the number off average and all cigarette consumption in the year 2010. Now, in the next question, what is the average daily cigarette consumption per smoker? So this ihsaa 4050 per capita, actually, as the information is there in the question No exclusion. Average daily cigarette consumption again, this is on par Capita, our pot smoker alert in the year 2010 is equal to the same value. This is for one year. That means 3 65 days, so divided by 3 65 will be called toe 11 point one cigarettes. But Governor Mark after not I can&#x27;t depart, smoke it. So is the same thing. So personal. There is smoking 11.1 cigarettes for today, which is a very huge number, and it&#x27;s highly contagious. I hope you want to shoot the calculation on explanation. Thank you.

Jodi Folley · Answer

for part A. We&#x27;re trying to find the percent of adults in the year 2005 using this formula and the X and the formula stands for the number of years after 1965. So we got a first figure out. Our X X is gonna be equal to 2005 minus 1965. So that&#x27;s 40 years. So we&#x27;re gonna plug that in for X so C equals negative quite five times 40 plus 41. So that equals negative 20 plus 41 and not equals 21 percent. And that matches the bar graph matches the graph, Part B. We want the percent in 2019. So X is going to be 2019 minus 1965. So that is 54 years. So now we&#x27;re gonna dio c equals negative 0.5 times 54 plus 41. So we get that, that&#x27;s gonna be negative. 27 plus 41. That leaves us with 14 and that&#x27;s a percent amount

Jaida L · Answer

So if we want to check how many people are smoking cigarettes in 2009 using the model, all we have to do is take a model that they give us and click in X. However, what is X? It&#x27;s not 2009 in the problem. It tells you that, um X resembles the number of years after 1965. So according to this model, what is X when the year is 2009? Well, if you can use your calculator, you can dio 2000 and nine minus 1965. So then you get 44. Which makes sense. Um, so 44 years after 1965. The years 2000 So x is not 2009 excess 44. Once you find X, you can plug that into the equation, so C equals negative 0.5 times 44 plus 41 then we could use our calculator. So negative 410.5 times 44 is negative 22 and then plus 41. So the final answer is 19 and C represents the percent Americans who smoked cigarettes. And so when the years 2000 and nine, according to the model 19% Americans smoked cigarettes. However, this equation that we just use is a model. According to the graph, the actual amount of purse the actual percent of Americans who smoked cigarettes in 2009 is 20%. So this is the rial data. This is the of 90% is the estimated data. So, as you can see, the model is a underestimate, a nest image of the actual percent of Americans who smoke cigarettes. And not only that, it is an underestimate by 1%. So we just saw the part A of question 1 19 So let&#x27;s look apart. Be if we want to use this same model that is given to estimate the percent of Americans who smoked cigarettes in the year 2015. Once again, we need to figure out what is X if the years 2015. So I&#x27;m want to do this problem in red. Sorry for the typo foods. What is exe when the year is 2015? Well, we have to do the same thing because X resembles the number of years after 1965 we need to take 2000 and 15 minus 1965 to figure out what X is not ends up being 50. So we know that access 50. So what we can do is take the original equation and plug and 50 for X Bus 41 we can use our calculator again to solve. So negative 410.5 times 50 is equal to negative 25 and then plus 41. And that whole expression ends up being 16. So using that model, the C equals using this model. The projected amount a percent of Americans who smoke who will smoke in 2015 ends up being 16%. So here are the two answers. This is the answer for part A, and this is the answer for party.

Joanna Quigley · Answer

So we have a mathematical model for the percentage of Americans who smoke cigarettes, where P is for cents and X is for number of years after 1970 funds. Most Wiggins pelvis speaker off in the textbook and look at the percentage of American adults who smoke in 20 cents. Eso here X equals 40 40 years after, So if we put this in, we get P. I know so ex slightly ex of it on I get he 17 17%. I&#x27;m looking at the Gruff are in 2010. It&#x27;s 19 cents, so it&#x27;s underestimates by 2% this high. Then we have a look at pots Be, which is using our formula to projects for year. Web scented is 7% so seven plus X Liberty course 37 we&#x27;re solving for X it so X over, too. Who&#x27;s 30 X equals 60 and this is 60 years after 1970. So our answer is 20 buddy

Joanna Quigley · Answer

so using our mathematical model for beeps sensitive American adults who smoke cigarettes. And first we want to have a look Have a model compares to be graph in being textbook for the year to thousands on X is a number of years after 1970. So x who was 30 and people percent p plus 30 over two. Who&#x27;s 37 S. O P. Whose 37 minus 15 Uh, so we&#x27;re looking at 22 years? Um, so this he&#x27;s going to give us 1%. It&#x27;s giving him a gruff cigar offices, 23%. So we have a one person underestimate competitive girl. That&#x27;s possible. But, Poppy, we&#x27;re going to use our muzzle to project forwards to the time where only 2% smoked cigarettes. So here p equals two, and we have two plus X over to cause 37. So x two I was 35. X equals 76 years after 1970 so our answer is going to be 2040

Problem

The sales $ y $ (in billions of dollars) for Harl…

Problem 85 Easy Difficulty

Answer

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

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

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

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Precalculus with Limits