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.

a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

Accepted Answer

Understand that the function $$V'(t) = 70(1 + \sin 2\pi t)$$ represents the rate of blood flow in milliliters per second at time $$t$$, and $$V(t) is $$the total volume pumped from time 0 to time $$t. To $$find the total amount of blood pumped over the interval from $$t=0 to$$ $$t=1$$ second, we need to integrate the rate function $$V'(t)$$ over this interval. This means calculating $$V(1) - V(0) = \$$int_0$$^1 V'(t) \, dt. $$Set up the integral: $$\$$int_0$$^1 70(1 + \sin 2\pi t) \, dt. $$This integral can be split into two parts: $$70 \$$int_0$$^1 1 \, dt + 70 \$$int_0$$^1 \sin 2\pi t \, dt. $$Evaluate each integral separately: the first integral $$\$$int_0$$^1 1 \, dt is $$straightforward, and the second integral $$\$$int_0$$^1 \sin 2\pi t \, dt$$ involves the sine function with a frequency of $$2\pi. $$Add the results of the two integrals and multiply by 70 to find the total volume pumped over one second. This value should verify that the amount of blood pumped in one second is 70 mL.

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Key Concepts

Derivative as Rate of Change

Definite Integral and Accumulated Quantity

Properties of the Sine Function in Modeling