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Chapter 8: Problem 3

In 1966 , the U.S. Surgeon General's health warnings began appearing on cigarette packages. Data gathered after 1965 indicate that public awareness of the health hazards of smoking has had some effect on consumption of cigarettes. The percentage, \(p\), of the total population ( 18 and older) who smoked \(t\) years after 1965 can be approximated by the model \(p=-0.555 t+41.67,\) where \(t\) is the number of years after 1965 Note that \(t=0\) corresponds to the year \(1965, t=9\) corresponds to \(1974,\) etc. a. Determine the percentage of the population who smoked in \(1981(t=16)\) b. Use the formula to determine the year when the percentage of smokers will be \(15 \%\) \((p=15) ?\)

Short Answer

Expert verified
Answer 1: 32.79% Question 2: In which year will the percentage of smokers in the total population be approximately 15%? Answer 2: 2013

Step by step solution

01

Calculate the Percentage in 1981 (t=16)

To calculate the percentage of the total population who smoked in 1981, replace t with 16 in the equation: $$p=-0.555(16)+41.67.$$ Now, we can calculate the percentage: $$p = -0.555(16)+41.67 = -8.88+41.67 = 32.79$$ So in 1981, approximately 32.79% of the total population smoked.
02

Determine the Year When Percentage of Smokers Will Be 15%

To find the year when the percentage of smokers will be 15%, set p = 15 and solve for t in the equation: $$15= -0.555t + 41.67.$$ Rearrange the equation to isolate t: $$0.555t =41.67 - 15,$$ $$t=\frac{41.67-15}{0.555} $$ Now, we can calculate the value of t: $$t=\frac{26.67}{0.555} = 48.06$$ Since t represents the number of years after 1965, we can determine the actual year by adding 1965: Year = 1965 + 48.06 鈮� 1965 + 48 = 2013 So, the percentage of smokers in the total population will be approximately 15% in the year 2013.

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