An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, \( P \), who contracted the flu \( t \) days after it brove out is given by the model \( P=-t^{2}+13 t+150 \), where \( 1 \leq t \leq 6 \). Find the day that 180 students had the flu. Recall that the restriction on \( t \) is at most 6 . \( t=\square \) Submilinging

To find the day when 180 students had the flu, we need to solve the equation \( P = -t^2 + 13t + 150 \) for \( t \) when \( P = 180 \). Given equation: \( P = -t^2 + 13t + 150 \) Substitute \( P = 180 \) into the equation: \[ 180 = -t^2 + 13t + 150 \] Rearrange the equation to standard quadratic form: \[ t^2 - 13t + 30 = 0 \] Now, we can solve this quadratic equation to find the value of \( t \) when 180 students had the flu. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(t^{2}-13t+30=0\) - step1: Factor the expression: \(\left(t-10\right)\left(t-3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&t-10=0\\&t-3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&t=10\\&t=3\end{align}\) - step4: Rewrite: \(t_{1}=3,t_{2}=10\) The solutions to the quadratic equation \( t^2 - 13t + 30 = 0 \) are \( t = 3 \) and \( t = 10 \). Since the restriction on \( t \) is at most 6, the day when 180 students had the flu is \( t = 3 \).