Question

1. the number of students at the ovhs football game after gates open is modeled by s(t) is the number of students and t is time in hours. $s(t)=\begin{cases}50t + 100, &0leq tleq4\\400, &4leq t<5\\frac{2500 + 100t}{5 + t}, &tgeq5end{cases}$ find $lim_{t

ightarrowinfty}s(t)$. explain what this means in the context of the problem. find $lim_{t
ightarrow4^{-}}s(t)$

Explanation:

Step1: Find $\lim_{t

ightarrow4^{-}}S(t)$
For $t
ightarrow4^{-}$, we use the first - piece of the piece - wise function $S(t)=50t + 100$ since $0\leq t\leq4$. Substitute $t = 4$ into $50t+100$. So, $S(t)=50\times4 + 100=200 + 100=300$.

Step2: Find $\lim_{t

ightarrow\infty}S(t)$
For $t\geq5$, $S(t)=\frac{2500 + 100t}{5 + t}$. Divide both the numerator and denominator by $t$: $S(t)=\frac{\frac{2500}{t}+100}{\frac{5}{t}+1}$. As $t
ightarrow\infty$, $\frac{2500}{t}
ightarrow0$ and $\frac{5}{t}
ightarrow0$. So, $\lim_{t
ightarrow\infty}S(t)=\frac{0 + 100}{0+1}=100$.

Answer:

$\lim_{t
ightarrow4^{-}}S(t)=300$, $\lim_{t
ightarrow\infty}S(t)=100$. In the context of the problem, $\lim_{t
ightarrow4^{-}}S(t)$ represents the number of students approaching the value as time gets closer to 4 hours (from the left - hand side, before the 4 - hour mark), which is 300 students. $\lim_{t
ightarrow\infty}S(t)$ represents the long - term number of students at the football game as time goes on indefinitely, which is 100 students.