Question

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.

Answer 1

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:

`f(t)=a(t-h)^(2)+k`

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

`f(t)=a(t-0)^(2)+49\\ f(t)=at^(2)+49`

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

`36=a(8)^(2)+49\Rightarrow 64a=-13\Rightarrow a=-(13)/(64)`

Therefore, the required quadratic model is
`f(t)=-(13)/(64)(t)^(2)+49`

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

`f(12)=-(13)/(64)(12)^(2)+49\\ \\ f(12)=-(13)/(64)(144)+49\\ \\ f(12)=-29.25+49=19.75`

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