Question

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 pequalsminus1.881tplus60.673​, where p is equal to the percent and t is equal to number of years after 2000. When will the percent be less than 27​%?
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Part 1
The percent will be less than 27​% after the year
  
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Answer 1

To determine when the percent of 12th graders who have ever used cigarettes will be less than 27%, we need to set up the equation according to the given model:

\[ p = -1.881t + 60.673 \]

We want to find the year \( t \) when \( p < 27 \).

Setting up the inequality:

\[ -1.881t + 60.673 < 27 \]

Now, let's solve for \( t \):

1. Subtract 60.673 from both sides:

\[ -1.881t < 27 - 60.673 \]

\[ -1.881t < -33.673 \]

2. Divide by -1.881 (Note that when dividing by a negative number, the inequality sign will flip):

\[ t > \frac{-33.673}{-1.881} \]

\[ t > 17.9 \]

Since \( t \) represents the number of years after 2000, we can round up 17.9 to the next whole number because time would have to pass fully into the next year. Thus:

\[ t = 18 \]

Now, we find the corresponding year:

\[ 2000 + 18 = 2018 \]

Thus, the percent will be less than 27% after the year \( \textbf{2018} \).