Questions: Suppose that Jessica's blood pressure can be modeled by the following function. p(t) = 92 + 20 cos(82 pi t) Jessica's blood pressure increases each time her heart beats, and it decreases as her heart rests in between beats. In this equation, p(t) is the blood pressure in mmHg (millimeters of mercury), and t is the time in minutes. Find the following. If necessary, round to the nearest hundredth. Number of heartbeats per minute: Maximum blood pressure: mmHg Period of p: minutes

Solution Steps

To solve the given problem, we need to analyze the function \( p(t) = 92 + 20 \cos(82 \pi t) \).

1. Number of heartbeats per minute: The number of heartbeats per minute corresponds to the frequency of the cosine function. The general form of a cosine function is \( a \cos(b t + c) + d \), where the frequency is given by \( \frac{b}{2\pi} \). Here, \( b = 82\pi \).

2. Maximum blood pressure: The maximum value of the cosine function is 1. Therefore, the maximum blood pressure is \( 92 + 20 \times 1 \).

3. Period of \( p \): The period of a cosine function is given by \( \frac{2\pi}{b} \). Here, \( b = 82\pi \).

The number of heartbeats per minute is calculated using the formula for frequency derived from the cosine function. Given \( b = 82\pi \), we find:

\[ \text{Heartbeats per minute} = \frac{b}{2\pi} = \frac{82\pi}{2\pi} = 41 \]

The maximum blood pressure occurs when the cosine function reaches its maximum value of 1. Thus, we calculate:

\[ \text{Maximum blood pressure} = 92 + 20 \cdot 1 = 112 \, \text{mmHg} \]

The period of the function \( p(t) \) is determined using the formula for the period of a cosine function:

\[ \text{Period of } p = \frac{2\pi}{b} = \frac{2\pi}{82\pi} = \frac{1}{41} \approx 0.0244 \, \text{minutes} \]

Final Answer