Video: Calculating the Change in Channel Dimensions given the Flow Rate Change

Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased to increase blood flow by a factor of 7.0?

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Video Transcript

Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what factor must the radius of an artery be increased to increase blood flow by a factor of 7.0?

In this situation, if we imagine that the artery initially has a radius π‘Ÿ sub one and after the operation has a radius π‘Ÿ sub two, then we want to figure out what is π‘Ÿ two divided by π‘Ÿ one. We can recall that, for a liquid, the flow rate of that liquid equals the cross-sectional area of the tube or pipe it passes through multiplied by the speed of the liquid. In this scenario, we can assume that the speed of the blood flow in the original artery is the same as the blood flow speed in the extended artery. This means we can write that 𝐴 two, the cross-sectional area of the dilated artery, times 𝑣 is equal to 7.0 times 𝐴 one times 𝑣. The fluid flow speed 𝑣 cancels out. And if we assume that the artery has a circular cross section, we can write that πœ‹ times π‘Ÿ two squared is equal to 7.0 times πœ‹ times π‘Ÿ one squared. We see the factors of πœ‹ cancel out.

If we divide both sides by π‘Ÿ one squared and take the square root of both sides, we see that π‘Ÿ two over π‘Ÿ one equals the square root of 7.0, or to two significant figures 2.6. That’s the factor by which the artery radius must be dilated in order to increase the flow rate by a factor of 7.0.

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