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 pequalsminus1.611tplus58.558​, 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​%?

Answer 1

To determine when the percent of 12th graders who have ever used cigarettes will be less than 21%, we need to solve the inequality given by the model:

\[
p = -1.611t + 58.558
\]

We want to find \(t\) such that \(p < 21\).

Substituting \(21\) into the equation:

\[
-1.611t + 58.558 < 21
\]

Now, we will isolate \(t\):

1. Subtract \(58.558\) from both sides:

\[
-1.611t < 21 - 58.558
\]
\[
-1.611t < -37.558
\]

2. Now, divide both sides by \(-1.611\). Remember that when dividing by a negative number, we must flip the inequality sign:

\[
t > \frac{-37.558}{-1.611}
\]
\[
t > 23.3
\]

Since \(t\) represents the number of years after 2000, we need to find the corresponding year:

\[
t > 23.3 \implies t \geq 24 \quad (\text{since } t \text{ must be a whole number})
\]

Calculating the year:

\[
2000 + 24 = 2024
\]

Therefore, the percent of 12th graders who have ever used cigarettes will be less than 21% after the year **2024**.