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Math  /  Algebra

QuestionQuestion 4 (1 point) The function below is a blood pressure model that describes a person's blood pressure in mm Hg as a function of time in seconds. P(t)=15sin⁡(13π6t)+110P(t)=15 \sin \left(\frac{13 \pi}{6} t\right)+110
What is this person's heart rate in beats per minute? A bpm What is this person's maximum blood pressure? □\square undefinedmmHg\underbrace{}_{\mathrm{mmHg}}
What is this person's minimum blood pressure? □\square A) mmHg

Studdy Solution

STEP 1

1. The function P(t)=15sin⁡(13π6t)+110 P(t) = 15 \sin \left(\frac{13 \pi}{6} t\right) + 110 models blood pressure as a sinusoidal function.
2. The heart rate in beats per minute (bpm) can be determined from the period of the sine function.
3. The maximum and minimum blood pressure values can be determined from the amplitude and vertical shift of the sine function.

STEP 2

1. Determine the period of the sine function to find the heart rate.
2. Calculate the maximum blood pressure.
3. Calculate the minimum blood pressure.

STEP 3

To find the heart rate, we need to determine the period of the sine function. The general form of a sine function is asin⁡(bt+c)+d a \sin(b t + c) + d , where the period is given by 2πb \frac{2\pi}{b} .
For the function P(t)=15sin⁡(13π6t)+110 P(t) = 15 \sin \left(\frac{13 \pi}{6} t\right) + 110 , the coefficient b b is 13π6 \frac{13 \pi}{6} .

STEP 4

Calculate the period using the formula:
Period=2π13π6=2π×613π=1213 \text{Period} = \frac{2\pi}{\frac{13\pi}{6}} = \frac{2\pi \times 6}{13\pi} = \frac{12}{13}
This period is in seconds per beat.

STEP 5

Convert the period from seconds per beat to beats per minute (bpm). There are 60 seconds in a minute, so:
Heart Rate (bpm)=601213=60×1312=65 \text{Heart Rate (bpm)} = \frac{60}{\frac{12}{13}} = 60 \times \frac{13}{12} = 65

STEP 6

The maximum blood pressure occurs at the maximum value of the sine function, which is 1. The function is:
P(t)=15sin⁡(13π6t)+110 P(t) = 15 \sin \left(\frac{13 \pi}{6} t\right) + 110
The maximum value of sin⁡(13π6t) \sin \left(\frac{13 \pi}{6} t\right) is 1, so:
Maximum Pressure=15×1+110=125 \text{Maximum Pressure} = 15 \times 1 + 110 = 125

STEP 7

The minimum blood pressure occurs at the minimum value of the sine function, which is -1. The function is:
P(t)=15sin⁡(13π6t)+110 P(t) = 15 \sin \left(\frac{13 \pi}{6} t\right) + 110
The minimum value of sin⁡(13π6t) \sin \left(\frac{13 \pi}{6} t\right) is -1, so:
Minimum Pressure=15×(−1)+110=95 \text{Minimum Pressure} = 15 \times (-1) + 110 = 95
The person's heart rate is 65 \boxed{65} bpm, the maximum blood pressure is 125 \boxed{125} mmHg, and the minimum blood pressure is 95 \boxed{95} mmHg.

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