Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


c. What is the cardiac output over a period of 1 min? (Use calculus; then check your answer with algebra.)

Verified step by step guidance

1. Identify the given rate of blood flow as the derivative of the volume function: \(V'(t) = 70(1 + \sin(2\pi t))\), where \(V(t)\) is the total volume pumped from time 0 to time \(t\) seconds.
2. To find the cardiac output over 1 minute, which is 60 seconds, set up the definite integral of the rate function from \(t=0\) to \(t=60\): \(\displaystyle V(60) = \int_0^{60} 70(1 + \sin(2\pi t)) \, dt\)
3. Split the integral into two parts to simplify the calculation: \(\displaystyle V(60) = 70 \int_0^{60} 1 \, dt + 70 \int_0^{60} \sin(2\pi t) \, dt\)
4. Evaluate each integral separately: - The integral of 1 with respect to \(t\) over \([0,60]\) is straightforward: \(\int_0^{60} 1 \, dt = 60\). - For the integral of \(\sin(2\pi t)\), use the antiderivative formula for sine: \(\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C\). Apply this to \(\int_0^{60} \sin(2\pi t) \, dt\).
5. Combine the results of the integrals and multiply by 70 to find \(V(60)\), which represents the total volume pumped in 60 seconds (1 minute). This gives the cardiac output over that period. Finally, verify your answer by considering the average flow rate algebraically and multiplying by the total time.

Key Concepts

Derivative and Rate of Change

The derivative represents the instantaneous rate of change of a function. In this problem, V′(t) models the rate at which blood is pumped at time t, showing how volume changes per second. Understanding derivatives helps interpret how the heart's outflow varies over time.

Definite Integral and Accumulated Quantity

The definite integral of a rate function over an interval gives the total accumulated quantity during that time. Here, integrating V′(t) from 0 to 60 seconds calculates the total blood volume pumped in one minute, linking rate of flow to total output.

Periodic Functions and Sinusoidal Models

Sinusoidal functions like sine model periodic phenomena such as heartbeats. The term sin(2πt) reflects the cyclical nature of blood flow with each beat, allowing the rate function to capture fluctuations around the average stroke volume.