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 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.

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.

Answer 1

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

1. Domain: The domain of the function is \( P \geq 0 \) because the number of patients cannot be negative.

2. Range: The range of \( T \) is \( T \geq 0 \), because the time a patient waits cannot be negative.

3. Minimum Point: When \( P = 0 \), \( T = 15\sqrt{0} = 0 \). This means when there are no patients in the waiting room, a patient doesn’t wait at all. Hence the minimum point is (0,0).

4. Increasing Function: The function is indeed increasing, but it increases at a decreasing rate. This is because the derivative of \( T \) with respect to \( P \) is \( \frac{dT}{dP} = \frac{15}{2\sqrt{P}} \), which decreases as \( P \) increases. This indicates that with each additional patient, the increase in wait time becomes smaller.

5. Constant Rate of Increase: The statement about the function increasing at a constant rate is incorrect; the function does not increase at a constant rate due to the square root relationship.

Based on this analysis, the correct interpretation from the provided responses is:

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

This option correctly describes the nature of the function.