At rest, the human heart beats once every second. At the strongest part of the
beat, a person's blood pressure peaks at 120mmHg. At the most relaxed part
of the beat, a person's blood pressure drops to 80mmHg. Supposing that t = 0
corresponds to the peak of the beat, write down a trigonometric function that
captures this behavior.
since it's a max at t=0, that indicates a cosine function
The period is 1, so we are looking at cos(2pi*t)
max is 120, min is 80, so amplitude is 20, and oscillates around h=100, and our function is
f(t) = 100 + 20cos(2pi*t)
To write a trigonometric function that captures the behavior of the human heart beat, we can use a sine function since it oscillates between maximum and minimum values.
Let's start by defining the period of the function, which is the time it takes for one complete heartbeat. In this case, the period is 1 second since the human heart beats once every second.
Since the function starts at the peak of the beat at t = 0, we can use the general form of a sine function to write the equation:
f(t) = A * sin(ωt + φ) + B
Where:
- A is the amplitude of the function, which corresponds to the difference between the maximum and minimum values of the blood pressure. In this case, A = (120 - 80) / 2 = 20 mmHg.
- ω is the angular frequency of the function, which is related to the period T by the formula ω = 2π / T. In this case, since T = 1 second, ω = 2π / 1 = 2π.
- φ is the phase shift, which determines the starting point of the function. In this case, since we want the function to start at t = 0, φ = 0.
- B is the vertical shift, which determines the average value of the function. In this case, B = (120 + 80) / 2 = 100 mmHg.
Putting it all together, the equation becomes:
f(t) = 20 * sin(2πt) + 100
Therefore, the trigonometric function that captures the behavior of the human heart beat is:
f(t) = 20 * sin(2πt) + 100 mmHg.