Answered

Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

(c). It is well known that the rate of flow can be found by measuring the volume of blood that flows past a point in a given time period. The volume V of blood flow through the blood vessel is V = =[₁2πvr dr where v= K (R² − r²) is the velocity of the blood through a vessel. In the velocity of the blood through a vessel, K is a constant, the maximum velocity of the blood, R is a constant, the radius of the blood vessel and r is the distance of the particular corpuscle from the center of the blood vessel.

(i) If R = 0.30 cm and v= (0.30-3.33r²) cm/s, find the volume.

(ii) Construct and develop a general formula for the volume of the blood flow. [Verify your answer by MATHEMATICA and attach the printout of the commands and output

C It Is Well Known That The Rate Of Flow Can Be Found By Measuring The Volume Of Blood That Flows Past A Point In A Given Time Period The Volume V Of Blood Flow class=

Sagot :

LammettHash

(i) Given that

[tex]V(R) = \displaystyle \int_0^R 2\pi K(R^2r-r^3) \, dr[/tex]

when R = 0.30 cm and v = (0.30 - 3.33r²) cm/s (which additionally tells us to take K = 1), then

[tex]V(0.30) = \displaystyle \int_0^{0.30} 2\pi \left(0.30-3.33r^2\right)r \, dr \approx \boxed{0.0425}[/tex]

and this is a volume so it must be reported with units of cm³.

In Mathematica, you can first define the velocity function with

v[r_] := 0.30 - 3.33r^2

and additionally define the volume function with

V[R_] := Integrate[2 Pi v[r] r, {r, 0, R}]

Then get the desired volume by running V[0.30].

(ii) In full, the volume function is

[tex]\displaystyle \int_0^R 2\pi K(R^2-r^2)r \, dr[/tex]

Compute the integral:

[tex]V(R) = \displaystyle \int_0^R 2\pi K(R^2-r^2)r \, dr[/tex]

[tex]V(R) = \displaystyle 2\pi K \int_0^R (R^2r-r^3) \, dr[/tex]

[tex]V(R) = \displaystyle 2\pi K \left(\frac12 R^2r^2 - \frac14 r^4\right)\bigg_0^R[/tex]

[tex]V(R) = \displaystyle 2\pi K \left(\frac{R^4}2- \frac{R^4}4\right) [/tex]

[tex]V(R) = \displaystyle \boxed{\frac{\pi KR^4}2} [/tex]

In M, redefine the velocity function as

v[r_] := k*(R^2 - r^2)

(you can't use capital K because it's reserved for a built-in function)

Then run

Integrate[2 Pi v[r] r, {r, 0, R}]

This may take a little longer to compute than expected because M tries to generate a result to cover all cases (it doesn't automatically know that R is a real number, for instance). You can make it run faster by including the Assumptions option, as with

Integrate[2 Pi v[r] r, {r, 0, R}, Assumptions -> R > 0]

which ensures that R is positive, and moreover a real number.

We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.