Question

1. the graph shows the rate at which people enter an art museum, a(t), measured in people per hour, where t is measured in hours after the museum opens. a. identify the interval(s) on which a is increasing. what does this mean in the context of this problem? b. identify the interval(s) on which a is constant. what does this mean in the context of this problem? c. identify the maximum of a and interpret its meaning in the context of this problem. d. when is a(t)=0? what does this mean in the context of this problem?

Explanation:

Step1: Analyze increasing interval

By observing the graph, the function $a(t)$ is increasing when the slope of the graph is positive. We can see that $a(t)$ is increasing on the interval $0 < t< 3$. In the context of the problem, it means that the rate at which people enter the art - museum is increasing in the first 3 hours after the museum opens.

Step2: Analyze constant interval

The function $a(t)$ is constant when the graph is a horizontal line. Here, $a(t)$ is constant on the interval $5 < t< 7$. In the context of the problem, it means that the rate at which people enter the art - museum is steady (constant number of people per hour) from 5 hours to 7 hours after the museum opens.

Step3: Find the maximum

The maximum value of $a(t)$ occurs at the highest point on the graph. The maximum value of $a(t)$ is 180 people per hour, which occurs at $t = 4$. In the context of the problem, it means that 4 hours after the museum opens, the rate at which people enter the art - museum is at its highest, with 180 people entering per hour.

Step4: Find when $a(t)=0$

We look for the points where the graph intersects the $t$-axis. From the graph, $a(t)=0$ at $t = 0$ and $t = 9$. At $t = 0$, it means that before the museum opens, the rate of people entering is 0. At $t = 9$, it means that 9 hours after the museum opens, no more people are entering the museum.

Answer:

a. The interval on which $a$ is increasing is $(0,3)$. It means the rate of people entering the museum is rising in the first 3 hours after it opens.
b. The interval on which $a$ is constant is $(5,7)$. It means the rate of people entering the museum is steady from 5 hours to 7 hours after it opens.
c. The maximum of $a$ is 180 people per hour at $t = 4$. It means 4 hours after the museum opens, the rate of people - entering is the highest at 180 people per hour.
d. $a(t)=0$ at $t = 0$ and $t = 9$. At $t = 0$, no one is entering as the museum has not opened yet. At $t = 9$, no more people are entering the museum.