Underage smoking. the number of underage cigarette smokers (ages 10–17) has declined in the united states. the peak percent was in 1998 at 49%. in 2006 this had dropped to 36%. let t be time in years after 1998 (t = 0 corresponds to 1998).
a. find a quadratic function that models the percent of underage smokers as a function of time. let (0, 49) be the vertex.
b. now that you have the model, predict the percent of underage smokers in 2010.

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ApusApus ApusApus · Answer 1 · 2018-09-12T11:50:43+02:00

Let f(t) be the percentage of underage cigarette smokers t years after 1998.

We are also given that percentage of underage cigarette smokers was 49% (highest) in year 1998.

(A) We are supposed to assume (0,49) as vertex of our model.

Let us assume that our quadratic model is:

[tex]f(t)=a(t-h)^{2}+k[/tex]

Upon substituting the vertex (h,k) = (0,49), we get:

[tex]f(t)=a(t-0)^{2}+49\\ f(t)=at^{2}+49[/tex]

We can find the value of 'a' using the fact that in year 2006 (t = 8) there were 36% underage smokers.

[tex]36=a(8)^{2}+49\Rightarrow 64a=-13\Rightarrow a=-\frac{13}{64}[/tex]

Therefore, the required quadratic model is [tex]f(t)=-\frac{13}{64}(t)^{2}+49[/tex]

(B) In order to predict the percentage of underage smokers in year 2010, we will substitute t=12 in our quadratic model.

[tex]f(12)=-\frac{13}{64}(12)^{2}+49\\ \\ f(12)=-\frac{13}{64}(144)+49\\ \\ f(12)=-29.25+49=19.75[/tex]

Therefore, there were 19.75% underage smokers in year 2010.