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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 pequalsminus1.841tplus59.334​, where p is equal to the percent and t is equal to number of years after 2000. When will the percent be less than 31​%?

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Part 1
The percent will be less than 31​% after the year
  
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1 answer

To find out when the percent \( p \) of 12th graders who have ever used cigarettes is less than 31%, we need to use the given model:

\[ p = -1.841t + 59.334 \]

We want to determine when \( p < 31 \). Setting up the inequality, we have:

\[ -1.841t + 59.334 < 31 \]

Now, we will solve this inequality step by step:

  1. Subtract 59.334 from both sides:

\[ -1.841t < 31 - 59.334 \] \[ -1.841t < -28.334 \]

  1. Divide both sides by -1.841. Remember, dividing by a negative number flips the inequality sign:

\[ t > \frac{-28.334}{-1.841} \]

  1. Calculating the right side:

\[ t > \frac{28.334}{1.841} \approx 15.4 \]

Since \( t \) represents the number of years after 2000, we round 15.4 up to the nearest whole number, which is 16.

This means that:

\[ t = 16 \]

This corresponds to the year 2000 + 16 = 2016.

So, the percent of 12th graders who have ever used cigarettes will be less than 31% after the year 2016.