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The percent of 12th graders in a certain state who have ever used cigarettes for each of the years 2000 through 2014 can be modeled by pequalsminus1.801tplus59.637​, where p is equal to the percent and t is equal to number of years after 2000. When will the percent be less than 21​%?

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Part 1
The percent will be less than 21​% after the year
  
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1 answer

To find the year when the percent of 12th graders who have ever used cigarettes will be less than 21%, we can start with the equation provided:

\[ p = -1.801t + 59.637 \]

We want to find the value of \( t \) when \( p < 21 \).

  1. Set up the inequality: \[ -1.801t + 59.637 < 21 \]

  2. Subtract 59.637 from both sides: \[ -1.801t < 21 - 59.637 \] \[ -1.801t < -38.637 \]

  3. Divide both sides by -1.801 (remember to flip the inequality sign): \[ t > \frac{-38.637}{-1.801} \] \[ t > 21.429 \]

Since \( t \) represents the number of years after 2000, we take the smallest integer greater than 21.429, which is 22.

  1. Now, to find the corresponding year: \[ 2000 + t = 2000 + 22 = 2022 \]

Thus, the percent of 12th graders who have ever used cigarettes will be less than 21% after the year 2022.