Suppose that a person’s blood pressure at time t in seconds is given by

p(t) = 100 + 18 sin (7t).
(a) During each heartbeat, what is the systolic pressure (maximum blood pressure)
and the diastolic pressure (minimum blood pressure)?
(b) According to this model, how many heartbeats are there per minute?

sin(anything) has a minimum of -1 and a max of +1

so 18sin(7t) has a min of -18 and a max of 18
since 18sin(7t) is added to 100
the minimum would be 100-18 or 82
the maximum would be 100+18 = 118

period of the function :
2π/7 = period
period = appr .897 sec per period
so in 1 minute we would have 60/.897 = appr 66.8 beats/minute

To find the systolic pressure (maximum blood pressure) and the diastolic pressure (minimum blood pressure), we need to identify the maximum and minimum values of the given function.

(a) To find the maximum and minimum values of the function p(t), we need to consider the function 18 sin(7t).

The maximum value of the sine function is 1, and the minimum value is -1. Therefore, the maximum and minimum values of 18 sin(7t) are 18 and -18, respectively.

Adding these values to the constant term 100, we can determine the systolic pressure (maximum blood pressure) and diastolic pressure (minimum blood pressure) as follows:

Systolic pressure (maximum blood pressure) = 100 + 18 = 118
Diastolic pressure (minimum blood pressure) = 100 - 18 = 82

(b) To determine the number of heartbeats per minute, we need to consider the frequency of the sine function. In this case, the frequency is 7.

Since one heartbeat corresponds to a complete cycle of the sine function, the number of heartbeats per second will be equal to the frequency, which is 7.

To convert the number of heartbeats per second to the number of heartbeats per minute, we multiply by 60 (the number of seconds in a minute).

Therefore, the number of heartbeats per minute is 7 * 60 = 420.

To find the systolic pressure (maximum blood pressure) and the diastolic pressure (minimum blood pressure) during each heartbeat, we need to understand the behavior of the given function, p(t) = 100 + 18 sin(7t).

(a) The given equation represents a sinusoidal function with a period of 2π/7, meaning it completes one full cycle in this interval. The maximum value of sin(7t) is 1, and the minimum value is -1.

The systolic pressure (maximum blood pressure) occurs at the peak of the function, which corresponds to the maximum value of sin(7t) = 1.
So, to find the systolic pressure, we substitute sin(7t) = 1 into the equation:
p(t) = 100 + 18(1) = 118 (mmHg)

The diastolic pressure (minimum blood pressure) occurs at the bottom of the function, which corresponds to the minimum value of sin(7t) = -1.
So, to find the diastolic pressure, we substitute sin(7t) = -1 into the equation:
p(t) = 100 + 18(-1) = 82 (mmHg)

Therefore, during each heartbeat, the systolic pressure is 118 mmHg, and the diastolic pressure is 82 mmHg.

(b) To determine the number of heartbeats per minute, we need to find the frequency of the function, which is the reciprocal of its period.

The period of sin(7t) is 2π/7. To convert this into minutes, we need to multiply by the appropriate factor:
1 minute = 60 seconds

So, the period in minutes is (2π/7) * (1/60) = π/210

The number of heartbeats per minute is the reciprocal of the period in minutes:
Number of heartbeats per minute = 1 / (π/210)
Number of heartbeats per minute = 210/π

Therefore, according to this model, there are 210/π (approximately 66.7) heartbeats per minute.