The average velocity of blood flowing in a certain 4 mm diameter artery in

the human body is 0.28 m/s. Calculate the Reynolds number and determine

whether the flow is laminar or turbulent. The viscosity and density of blood are

approximately 4 cP and 1.06 Mg/m3

, respectively.

P6.16

(a) Determine the volumetric flow rate of blood in the artery of P6.15.

(b) Calculate the maximum velocity of blood across the artery's cross section.

(c) Determine the amount by which the blood pressure decreases along each

10 cm length of the artery.

P6.17

The Boeing 787 Dreamliner is designed to be 20% more fuel efficient than the

comparable Boeing 767 and flies at an average cruise speed of Mach 0.85. The

midsize Boeing 767 has a range of 12,000 km, a fuel capacity of 90,000 L, and flies

at Mach 0.80. Assume the speed of sound is 700 mph, and calculate the projected

volumetric flow rate of fuel for each of the two Dreamliner engines in m3

/s.

P6.18

Assume that the fuel lines for each engine in the Boeing Dreamliner of P6.17

are 7/8 in. in diameter and that the density and viscosity of jet fuel are 800 kg/

m3

and 8.0 10?3

kg (m s), respectively. Calculate the average velocity of the

fuel in m/s and the Reynolds number of this flow. Also determine whether the

flow is laminar or turbulent.

P6.19

(a) For a 1.25 in. diameter pipe, what is the maximum volumetric flow rate

at which water can be pumped and the flow will remain laminar? Express

your result in the dimensions of gallons per minute.

(b) What would be the maximum flow rate for SAE 30 oil?

P6.20

At any time, approximately 20 volcanoes are actively erupting on the Earth, and

50-70 volcanoes erupt each year. Over the past 100 years, an average of 850

people have died each year from volcano eruptions. As scientists and engineers

study the mechanics of lava flow, accurately predicting the flow rate (velocity) of

the lava is critical to saving lives after an eruption. Jeffrey's equation captures

the relationship between flow rate and viscosity as:

V 5 rgt2 sin (a)

3m

1. Let (Y1, Y2, ..., YK) ~ Multinomial(n, 71, 72, ..., Wk), then (a) Calculate the moment generating function (This will be a multivariate MGF). [3 Marks] (b) Show that E(Y;) = naj 2 Marks (c) Show that Var(Y;) = naj (1 - nj) [4 Marks (d) Show that Cov(Yi, Y;) = -niT; [6 Marks] (e) Let's assume Yl, Y2, ..., Yo ~ Multinomial(n, 71, 72, ..., "C). Then show that the correla- tion coefficient between Yi and Yj is, Cor (Yi, Yj) = Vmi(1 - min;(1 -wj) [3 Marks (f) Show that for C = 2 the Cor(Y1, Y2) = -1. Explain why? [2 Marks Hint: You can use the Multivariate MGF. Recall if Y = (Y1, Y2, ..., Yk) be a multivariate random variable. Then the MGF is defined as, MY(t1, ..., th) = E(exp(t1 Y1 + ... + th Yk) ) Now use partial derivative on t; or to, t; to achieve the results1. Let (Y1, Y2, ..., YK) ~ Multinomial(n, 71, 72, ..., Wk), then (a) Calculate the moment generating function (This will be a multivariate MGF). [3 Marks] (b) Show that E(Y;) = naj [2 Marks] (c) Show that Var(Y;) = nn; (1 - ;) [4 Marks] (d) Show that Cov(Yi, Yj) = -ninj [6 Marks] Hint: You can use the Multivariate MGF. Recall if Y = (Y1, Y2, ..., Yk) be a multivariate random variable. Then the MGF is defined as, MY(t1, ..., tk) = E(exp(t1 1 + ... + thYk)) Now use partial derivative on t; or to, t; to achieve the results1. Recall the covariance of two random variables X and Y is defined as Cov(X, Y) = E[(X - E[X])(Y - E[Y])]. For a multivariate random variable Z (i.e., each index of Z is a random variable), we define the covariance matrix > such that E;; = Cov(Z;, Z;). Concisely, E = E[(Z -M)(Z -M) ], where u is the mean value of the random column vector Z. Prove that the covariance matrix is always positive semidefinite (PSD). Hint: Use linearity of expectation.known as a 10. A random variable that may take on any value in an interval or collection of intervals is a. continuous random variable. b. discrete random variable. c. mixed type random variable. d. multivariate random variable. e. none of the above 11. Which of the following is a required condition for a discrete probability function? a. [f(x) =0 for all values of x b. f(x) 21 for all values of x c. f(x)