The formula T=15P−−√

Question

can be used to estimate the time in minutes a patient waits in a waiting room at a doctor’s office given P patients in the waiting room.

What do the key features of the function reveal?

(1 point)
Responses

The range is T≤0
, so patients only wait negative amounts of time in the waiting room.
The range is cap t is less than or equal to 0 , so patients only wait negative amounts of time in the waiting room.

The minimum point (0,0)
indicates that when the number of patients in the waiting room is 0, the time they wait is 10 seconds.
The minimum point open paren 0 comma 0 close paren indicates that when the number of patients in the waiting room is 0, the time they wait is 10 seconds.

The function is an increasing function, so as the number of patients in the waiting room increases, the time they wait increases at a constant rate.
The function is an increasing function, so as the number of patients in the waiting room increases, the time they wait increases at a constant rate.

The function is an increasing function, so the time a patient waits in the waiting room is increasing at a decreasing rate.
The function is an increasing function, so the time a patient waits in the waiting room is increasing at a decreasing rate.

Answer 1

To analyze the function \( T = 15\sqrt{P} \), we can look at its key features:

1. Range and Values:

   • Since \( T \) (the wait time) is a function of the square root of \( P \) (the number of patients), \( P \) must be greater than or equal to 0. As a result, \( T \) will also be greater than or equal to 0 (i.e., \( T \geq 0 \)). This means that the function does not produce negative values; in fact, it shows that patients will wait a minimum of 0 minutes when there are 0 patients.

2. Minimum Point:

   • The minimum point is \( (0, 0) \). This indicates that when there are zero patients in the waiting room, the wait time is indeed zero minutes, not 10 seconds as suggested in the response options.

3. Increasing Function:

   • The function \( T = 15\sqrt{P} \) is an increasing function because the square root of \( P \) increases as \( P \) increases. However, it does so at a decreasing rate. As \( P \) increases, \( T \) increases, but the rate of increase diminishes (the slope of the function decreases).

Given these observations, the best choice from the responses provided is:

The function is an increasing function, so as the number of patients in the waiting room increases, the time they wait increases at a decreasing rate.

This accurately describes the nature of the function based on the analysis.