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 pequalsminus1.768tplus59.158​, where p is equal to the percent and t is equal to number of years after 2000. When will the percent be less than 26​%?

Question

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Part 1
The percent will be less than 26​% after the year
  
enter your response here.
​(Round to the nearest year as​ needed.)

Answer 1

To find out when the percent of 12th graders who have ever used cigarettes will be less than 26%, we can use the given model:

\[ p = -1.768t + 59.158 \]

We need to find when \( p < 26 \). Setting up the inequality:

\[ -1.768t + 59.158 < 26 \]

Now, subtract 59.158 from both sides:

\[ -1.768t < 26 - 59.158 \]

Calculating the right side:

\[ 26 - 59.158 = -33.158 \]

Now we have:

\[ -1.768t < -33.158 \]

To isolate \( t \), divide both sides by -1.768 (note that the direction of the inequality will change because we are dividing by a negative number):

\[ t > \frac{-33.158}{-1.768} \]

Calculating the division:

\[ t > 18.75 \]

Since \( t \) represents the number of years after 2000, we round 18.75 to the nearest whole number. Because \( t \) must be greater than 18.75, we round up to 19.

Now, adding 19 to the base year (2000):

\[ 2000 + 19 = 2019 \]

Thus, the percent of 12th graders who have ever used cigarettes will be less than 26% after the year 2019.