n 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=0.55t + 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 1991 (t=16)
b. Use the formula to determine the year when the percentage of smokers will be 15% (p=15)?
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To determine the percentage of the population who smoked in 1991 (t=16), we can substitute t=16 into the given equation: p = 0.55t + 41.67.
a. Substituting t=16, we have:
p = 0.55(16) + 41.67
p = 8.8 + 41.67
p ≈ 50.47
Therefore, approximately 50.47% of the population smoked in 1991.
b. To determine the year when the percentage of smokers will be 15% (p=15), we can rearrange the equation and solve for t.
p = 0.55t + 41.67
Let's substitute p=15 into the equation and solve for t:
15 = 0.55t + 41.67
Subtract 41.67 from both sides of the equation:
15 - 41.67 = 0.55t
-26.67 = 0.55t
Divide both sides by 0.55 to isolate t:
t ≈ -26.67 / 0.55
t ≈ -48.49
Keep in mind that t represents the number of years after 1965, so we need to subtract t from 1965 to get the year:
Year = 1965 - 48.49 ≈ 1916.51
Therefore, the formula predicts that the year when the percentage of smokers will be 15% is approximately 1916.51.