Chapter 7.3, Problem 103AYU

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 mmHg and the diastolic blood pressure is 80 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

$P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$

a. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is 100 mmHg.

b. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is 120 mmHg.

c. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is between 100 and 105 mmHg.

To determine

To find:

a. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is 100 mmHg.

Answer to Problem 103AYU

a. $t=0s,0.43s,0.86s$

Explanation of Solution

Given:

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 mmHg and the diastolic blood pressure is 80 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 $P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$.

Calculation:

a. Determine the times at which the blood pressure is 100 mmHg.

$P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$

$100=100+20\sin \left(\frac{7\pi }{3}t\right)$

$20\sin \left(\frac{7\pi }{3}t\right)=0$

$\sin \left(\frac{7\pi }{3}t\right)=0$

$\frac{7\pi }{3}t=0,\pi ,2\pi$

$t=0s,0.43s,0.86s$

To determine

To find:

b. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is 120 mmHg.

Answer to Problem 103AYU

b. $0.21s$

Explanation of Solution

Given:

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 mmHg and the diastolic blood pressure is 80 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 $P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$.

Calculation:

b. Determine the times at which the blood pressure is 120 mmHg.

$P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$

$120=100+20\sin \left(\frac{7\pi }{3}t\right)$

$20\sin \left(\frac{7\pi }{3}t\right)=20$

$\sin \left(\frac{7\pi }{3}t\right)=1$

$\frac{7\pi }{3}t=\frac{\pi }{2}$

$t=\frac{3}{14}=0.21s$

To determine

To find:

c. In the interval $\left[0,1\right]$, determine the times at which the blood pressure is between 100 and 105 mmHg.

Answer to Problem 103AYU

c. $\left[0,0.03\right]\cup \left[0.39,0.43\right]\cup \left[0.86\cup 0.89\right]$

Explanation of Solution

Given:

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 mmHg and the diastolic blood pressure is 80 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 $P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$.

Calculation:

c. Determine the times at which the blood pressure is between 100 and 105 mmHg.

The times at which the blood pressure is 100 mmHg is $t=0s,0.43s,0.86s$.

The times at which the blood pressure is 105 mmHg.

$P\left(t\right)=100+20\sin \left(\frac{7\pi }{3}t\right)$

$105=100+20\sin \left(\frac{7\pi }{3}t\right)$

$20\sin \left(\frac{7\pi }{3}t\right)=5$

$\sin \left(\frac{7\pi }{3}t\right)=\frac{1}{4}$

$t=0.07\pi ,0.7\pi ,2.08\pi$

$t=0.03s,0.39s,0.89s$

The times at which the blood pressure is between 100 and 105 mmHg.

$\left[0,0.03\right]\cup \left[0.39,0.43\right]\cup \left[0.86\cup 0.89\right]$