Blood Pressure Blood pressure is a way of measuring the...

Blood Pressure Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is $120 / 80$, which means the systolic blood pressure is $120 \mathrm{mmI}$ I and the diastolic blood pressure is $80 \mathrm{mmHg}$. Assuming that a person's heart beats 70 times per minute, the blood pressure $P$ of an individual after $t$ seconds can be modeled by the function $$ P(t)=100+20 \sin \left(\frac{7 \pi}{3} t\right) $$ (a) In the interval $[0,1]$, determine the times at which the blood pressure is $100 \mathrm{mmHg}$. (b) In the interval $[0,1]$, determine the times at which the blood pressure is $120 \mathrm{mmHg}$. (c) In the interval $[0,1]$, determine the times at which the blood pressure is between 100 and $105 \mathrm{mmHg}$.

Blood Pressure Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is $120 / 80$, which means the systolic blood pressure is $120 \mathrm{mmI}$ I and the diastolic blood pressure is $80 \mathrm{mmHg}$. Assuming that a person's heart beats 70 times per minute, the blood pressure $P$ of an individual after $t$ seconds can be modeled by the function
$$
P(t)=100+20 \sin \left(\frac{7 \pi}{3} t\right)
$$
(a) In the interval $[0,1]$, determine the times at which the blood pressure is $100 \mathrm{mmHg}$.
(b) In the interval $[0,1]$, determine the times at which the blood pressure is $120 \mathrm{mmHg}$.
(c) In the interval $[0,1]$, determine the times at which the blood pressure is between 100 and $105 \mathrm{mmHg}$.

Key Concepts

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, are used to model periodic phenomena. These functions are defined by parameters including amplitude, period, and vertical shift, which describe the oscillation’s intensity, frequency, and baseline value respectively. In this context, the blood pressure model is a sinusoidal function that represents the cyclic pattern of a beating heart over time.

Amplitude, Period, and Vertical Shift

The amplitude is the maximum deviation from the mean value (vertical shift) of a sinusoidal function, dictating the range of oscillation. The period is the time taken to complete one full cycle, calculated from the coefficient of the variable within the argument of the sine function. Understanding these parameters is crucial to analyzing the behavior of a system being modeled, such as variations in blood pressure during heartbeats.

Trigonometric Equation Solving

Solving trigonometric equations involves finding all angle measures that satisfy an equation like sin(?) = c. This requires the use of inverse trigonometric functions and understanding the periodic nature of sine. The general solution is typically expressed in terms of a base solution plus an integer multiple of the full period of the trigonometric function, a technique that is essential in determining multiple time instances that satisfy given blood pressure conditions.

Domain Restriction in Periodic Models

When modeling real-world phenomena, solutions must be found within specific intervals or domains that are physically meaningful. In this problem, the time interval is restricted, which means that only the solutions within this interval are relevant. This approach helps in filtering out extraneous solutions that do not correspond to the actual time frame being considered.

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