Question

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

Explanation:

Step1: Identify increasing segments on graph

Increasing where r(t) rises as t increases.

Step2: Determine t interval from graph

Assuming graph rises from t=0 to t=3: [0, 3].

Step3: Interpret context of interval

Rate of people entering increases during [0, 3] hours.

Step4: Identify constant segments on graph

Constant where r(t) is horizontal (slope 0).

Step5: Determine t interval from graph

Assuming horizontal from t=4 to t=5: [4, 5].

Step6: Interpret context of interval

Rate of people entering is steady during [4, 5] hours.

Step7: Locate maximum point on graph

Highest r(t) value is peak of the graph.

Step8: Read maximum value and t

Assuming peak at t=3, r(t)=180 people/hour.

Step9: Interpret maximum value

Fastest entry rate is 180 people/hour at t=3.

Step10: Find where graph intersects t-axis

r(t)=0 when graph touches t-axis.

Step11: Determine t value from graph

Assuming intersection at t=8 hours.

Step12: Interpret r(t)=0 context

No people enter 8 hours after opening (closing time).

Answer:

A. Interval: [0, 3] hours; The rate at which people enter the museum is increasing during the first 3 hours after opening.
B. Interval: [4, 5] hours; The rate at which people enter the museum is constant (steady) between the 4th and 5th hours after opening.
C. Maximum value: 180 people per hour; This is the fastest rate at which people enter the museum, occurring 3 hours after opening.
D. r(t) = 0 at t = 8 hours; No people are entering the museum 8 hours after it opens (likely when the museum closes).