Question

the velocity v of blood at a valve in the heart of a certain animal is modeled by the equation v = - 2 cos (4πt) + 2, where v is in centimeters per second and t is time in seconds. a. what are the maximum and minimum velocities of the blood at this valve? b. what is the animals heart rate in beats per minute? a. the maximum velocity of the blood at this valve is cm/sec and the minimum velocity of the blood at this valve is cm/sec.

Explanation:

Step1: Analyze the range of the cosine function

The range of the cosine function \(y = \cos(x)\) is \([- 1,1]\). For the function \(v=-2\cos(4\pi t)+2\), when \(\cos(4\pi t)=-1\):
\(v=-2\times(-1)+2\)
\(v = 2 + 2=4\)
When \(\cos(4\pi t)=1\):
\(v=-2\times1 + 2\)
\(v=0\)

Step2: Find the period of the function

The general form of a cosine function is \(y = A\cos(Bx - C)+D\), and its period is \(T=\frac{2\pi}{|B|}\). For the function \(v=-2\cos(4\pi t)+2\), \(B = 4\pi\). So the period \(T=\frac{2\pi}{4\pi}=\frac{1}{2}\) seconds.

Step3: Calculate the heart - rate

The heart - rate is the number of beats per minute. Since the period \(T=\frac{1}{2}\) seconds per beat. The number of beats per second is \(f=\frac{1}{T}\). So \(f = 2\) beats per second. To convert to beats per minute, we use the conversion factor: \(2\times60=120\) beats per minute.

Answer:

a. The maximum velocity of the blood at this valve is \(4\) cm/sec and the minimum velocity of the blood at this valve is \(0\) cm/sec.
b. The animal's heart rate is \(120\) beats per minute.