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Prachi, the office manager at a private practice, modeled a few relationships.

[tex]\[
\begin{tabular}{|l|l|l|}
\hline
Function & Input & Output \\
\hline
$P$ & Time of day, $t$ & People in waiting room, $P(t)$ \\
$W$ & People in waiting room, $n$ & Patient's waiting time, $W(n)$ \\
$L$ & People in waiting room, $x$ & Average length of visit, $L(x)$ \\
\hline
\end{tabular}
\][/tex]

The expression [tex]$L(P(t))$[/tex] represents which of the following?

Choose one answer:

A. A patient's waiting time as a function of the time of day

B. The average length of a patient visit as a function of the time of day

C. The number of people in the waiting room as a function of the time of day

Sagot :

GinnyAnsweridn
To solve this problem, let's understand the given functions and the relationships they describe:

1. Function [tex]\( P \)[/tex] (Time of day, [tex]\( t \)[/tex] → People in waiting room, [tex]\( P(t) \)[/tex]):
This function indicates that the number of people in the waiting room is dependent on the time of day. Specifically, given a specific time [tex]\( t \)[/tex], [tex]\( P(t) \)[/tex] tells us how many people are in the waiting room at that time.

2. Function [tex]\( W \)[/tex] (People in waiting room, [tex]\( n \)[/tex] → Patient's waiting time, [tex]\( W(n) \)[/tex]):
This function shows that the waiting time for a patient is dependent on the number of people in the waiting room. So, given [tex]\( n \)[/tex] people in the waiting room, [tex]\( W(n) \)[/tex] denotes the patient's waiting time.

3. Function [tex]\( L \)[/tex] (People in waiting room, [tex]\( x \)[/tex] → Average length of visit, [tex]\( L(x) \)[/tex]):
This function indicates that the average length of a patient's visit is dependent on the number of people in the waiting room. Specifically, for [tex]\( x \)[/tex] people in the waiting room, [tex]\( L(x) \)[/tex] gives the average length of a patient visit.

Now, let's analyze the expression [tex]\( L(P(t)) \)[/tex]:

1. [tex]\( P(t) \)[/tex] gives us the number of people [tex]\( n \)[/tex] in the waiting room at a specific time [tex]\( t \)[/tex].
2. [tex]\( L(n) \)[/tex] (or [tex]\( L(P(t)) \)[/tex] when [tex]\( n = P(t) \)[/tex]) then takes this number of people [tex]\( P(t) \)[/tex] as its input and provides the average length of a patient visit for that number of people.

Putting these relationships together:
- [tex]\( t \)[/tex] → [tex]\( P(t) \)[/tex] (time of day to number of people in waiting room)
- [tex]\( P(t) \)[/tex] → [tex]\( L(P(t)) \)[/tex] (number of people in waiting room to average length of visit)

Therefore, the expression [tex]\( L(P(t)) \)[/tex] provides the average length of a patient visit as it varies depending on the time of day.

So, the correct answer is:
(B) The average length of a patient visit as a function of the time of day
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