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Consider a one-dimensional plane wall with constant properties and uniform internal generation q. The left face is insulated, and the right face is held at a uniform temperature. T (a) Using the appropriate form of the heat equation. derive an expression for the x-dependence of the steady-state heat flux q"(x). (b) Using a finite volume spanning the range 0 S x 5 {, derive an expression for q"({) and compare the expression to your result for part (a).The steady-state temperature distribution in a semi- transparent material of thermal conductivity & and thickness L exposed to laser irradiation is of the form T(x) = - 4 e- + Bx + C ka where A, a, B. and C are known constants. For this situ- ation, radiation absorption in the material is manifested by a distributed heat generation term, q(x). Laser irradiation -L Semitransparent medium, T(x) (a) Obtain expressions for the conduction heat fluxes at the front and rear surfaces. (b) Derive an expression for q(x). (c) Derive an expression for the rate at which radiation is absorbed in the entire material, per unit surfaceFrom a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls. A typical roulette wheel used in a casino has 38 slots that are numbered 1, 2, 3, . . ., 36, 0, 00, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of 1/38, and we are interested in the number of the slot into which the ball falls. (a) Define the sample space S. (b) Let A = {0, 00). Give the value of P (A). (c) Let B = {14, 15, 17, 18). Give the value of P (B). (d) Let D = {x : x is odd). Give the value of P (D).Three cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades. Find a formula for the probability distribution of the random variable X representing the outcome when a single die is rolled once. Find the probability distribution of the random variable Win Exercise 3.3, assuming that the coin is biased so that a head is twice as likely to occur as a tail. Reference: Exercise 3.3: Let Wbe a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W.Construct a graph of the cumulative distribution function of Exercise 3. 15. Reference: Exercise 3. 15: Find the cumulative distribution function of the random variable X representing the number of defectives in Exercise 3.11. Then using F(x), find (a) P(X = 1); (b) P(0 3); (c) P(1.4 2)