Question

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.)

Accepted Answer

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. 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$$ Split the integral into two parts to simplify the calculation:

$$\displaystyle V(60) = 70 \$$int_0$$^{60} 1 \, dt + 70 \$$int_0$$^{60} \sin(2\pi t) \, dt$$ 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$$^{60} \sin(2\pi t) \, dt. $$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.

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Key Concepts

Derivative and Rate of Change

Definite Integral and Accumulated Quantity

Periodic Functions and Sinusoidal Models