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 pequalsminus1.678tplus59.859​, where p is equal to the percent and t is equal to number of years after 2000. When will the percent be less than 24​%?

Answer 1

To solve for when the percent \( p \) of 12th graders who have ever used cigarettes will be less than 24%, we begin with the given equation:

\[ p = -1.678t + 59.859 \]

We want to find \( t \) such that:

\[ p < 24 \]

Substituting \( p \) with 24 in the equation gives:

\[ -1.678t + 59.859 < 24 \]

Now, we can solve the inequality. First, let's isolate \( t \):

1. Subtract 59.859 from both sides:

\[ -1.678t < 24 - 59.859 \]

Calculating the right-hand side:

\[ 24 - 59.859 = -35.859 \]

Now the inequality is:

\[ -1.678t < -35.859 \]

2. Divide both sides by -1.678 (remember to reverse the inequality sign when dividing by a negative):

\[ t > \frac{-35.859}{-1.678} \]

Calculating the right-hand side:

\[ t > 21.4 \]

Since \( t \) represents the number of years after 2000, we need to round up to the nearest whole number because \( t \) must be an integer representing complete years. Thus, we will round up from 21.4:

\[ t \geq 22 \]

This means that \( t = 22 \) corresponds to the year:

\[ 2000 + 22 = 2022 \]

Therefore, the percent of 12th graders who have ever used cigarettes will be less than 24% starting in the year 2022.