Chapter 5.3, Problem 79E

Blood Pressure Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120/80 is considered normal.

A certain person’s blood pressure is modeled by the function

$p(t)=115+25\sin (160\pi t)$

where p(t) is the pressure in mmHg (millimeters of mercury), at time t measured in minutes.

1. (a) Find the period of p.
2. (b) Find the number of heartbeats per minute.
3. (c) Graph the function p.
4. (d) Find the blood pressure reading. How does this compare to normal blood pressure?

To determine

To find: The period of p.

Answer to Problem 79E

The period of the function $p\left(t\right)=115+25\sin \left(160\pi t\right)$ is $\frac{1}{80}\text{ minute}$.

Explanation of Solution

Given:

The function $p\left(t\right)=115+25\sin \left(160\pi t\right)$.

Sine curve:

The sine curve $y=a\sin kx\left(k>0\right)$ have amplitude $\left|a\right|$ and period $\frac{2\pi }{k}$.

Calculation:

The period of the function $p\left(t\right)=115+25\sin \left(160\pi t\right)$ is computed as follows,

$\begin{array}{c}\frac{2\pi }{k}=\frac{2\pi }{160\pi }\left(k=160\pi \right)\\ =\frac{1}{80}\end{array}$

Therefore, the period of the function $p\left(t\right)=115+25\sin \left(160\pi t\right)$ is $\frac{1}{80}\text{ minute}$.

To determine

To find: The number of heartbeats per minute.

Answer to Problem 79E

The number of heartbeats is 80 per minute.

Explanation of Solution

Calculation:

The period of the function $p\left(t\right)=115+25\sin \left(160\pi t\right)$ is $\frac{1}{80}\text{ minute}$.

The number of heartbeats per minute is computed as follows,

$\begin{array}{c}T=\frac{1}{p}\\ =\frac{1}{\frac{1}{80}}\\ =80\end{array}$

Therefore, the number of heartbeats is 80 per minute.

To determine

To sketch: The graph of the function p.

Explanation of Solution

Use the online graphing calculator and draw the graph of the function $p\left(t\right)=115+25\sin \left(160\pi t\right)$ as shown in below Figure 1.

From Figure 1, it is observed that the maximum is 140 and the minimum is 90. The range of the function is $\left[90,140\right]$.

To determine

To find: The blood pressure reading and compare to normal blood pressure.

Answer to Problem 79E

The required blood pressure $\frac{140}{90}$ is higher than the normal blood pressure.

Explanation of Solution

Calculation:

From Figure 1, it is observed that the maximum (systolic) is 140 and the minimum (diastolic) is 90.

$\begin{array}{c}\text{blood pressure}=\frac{\text{systolic}}{\text{diastolic}}\\ =\frac{140}{90}\end{array}$

That is, the required blood pressure reading is $\frac{140}{90}$.

Given that the normal blood pressure reading is $\frac{120}{80}$.

$\frac{140}{90}>\frac{120}{80}$

Hence, the required blood pressure $\frac{140}{90}$ is higher than the normal blood pressure.