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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

If on average one in twenty of a certain type of column will fail under a given axial load, what are the probabilities that among sixteen such columns, (a) At most two, (b) At least four will fail?
Assume that(a) An aircraft can land safely if at least half of its engines are working,(b) The probability of an engine failing is 0.1,(c) Engine failures are independent. Which is safer, a four-engine aircraft or a two engine aircraft?
If Z is a random variable having the standard normal distribution, find the probabilities that Z will have a value(a) Greater than 1.14,(b) Less than – 0.36,(c) Between – 0.46 and – 0.09,(d) Between – 0.58 and 1.12.
The probability of issuing a drill of high brittleness (a reject) is 0.02. Drills are packed in boxes of 100 each. What is the probability that the number of defective drills is no greater than two?
A town has five fire engines operating independently, each of which spends 94% of the time in its station awaiting a call. Find the probability that at least three fire engines are available when needed.
Eight babies are born in a hospital on a particular day. Find the probability that exactly half of them are boys. (The probability that a baby is a boy is actually slightly greater than one-half, but you can take it as exactly one-half for this exercise.)
Sample values that are several standard deviations away from the sample average are called outliers. They are often just measurement or transcription errors, but they can bias a statistical calculation. Which of the following data are more than three sample standard deviations away from the
Two people are separately attempting to succeed at a particular task, and each will continue attempting until success is achieved. The probability of success of each attempt for person A is p, and that for person B is q, all attempts being independent. What is the probability that person B will
Find the median and the mode for the Rayleigh distribution(see Question 31 in Exercises 13.4.5). Also show that the mean is given bywhich can be shown to be √(1/2πa). Compare these quantities when a = 6, and find the interquartile range.Data from Question 31The wave amplitude X on the sea
In a problem similar to that in Question 35 the probability of success at the first attempt is 0.2 but the probability of failure at each subsequent attempt (if needed) is half of that for the previous attempt. Find the mean number of attempts needed to achieve the first success.Data from Question
Find the sample averages and standard deviations for the engine performance data in Figure 13.1.Figure 13.1 Engine age data. Engine age 10 years Count 25 40 35 Defective 259
Find the sample average, standard deviation, median and range for the following sample of component lifetimes (in thousands of hours): 5.6, 4.1, 6.0, 5.8, 5.2, 4.3, 6.4, 5.5, 6.0, 5.1, 4.9, 4.2, 4.8, 6.8, 5.6, 5.2, 7.3, 5.4, 4.7, 5.9, 5.0, 6.3, 4.4, 6.0
An inspection of twelve specimens of material from inside a reactor vessel revealed the following percentages of impurities:Find (a) The sample average and both versions of the sample standard deviation, (b) The sample median and range. 2.3, 1.9, 2.1, 2.8, 2.3, 3.6, 1.4, 1.8, 2.1, 3.2, 2.0, 1.9
The mean times for completion of tasks A and B are four and six hours respectively. A particular project involves three tasks of type A and two of type B, all to be performed in succession. What is the expected time for completion of the project? Also, if the standard deviations for A and B are one
The distribution of downtime T for breakdowns of a computer system is given bywhere a is a positive constant. The cost of downtime derived from the disruption resulting from breakdowns rises exponentially with T: cost factor = h(T) = ebT Show that the expected cost factor for downtime is [a/(a –
If the probability density of the random variable X isfind the probability that X will take a value within two standard deviations of its mean. fx(x)= [30x(1-x) 0 (0 < x < 1) (otherwise)
Suppose that the running distance (in thousands of kilometres) that car owners get from a tyre is a random variable with density functionFind(a) The probability that one of these tyres will last at most 19 000 km;(b) The mean and standard deviation of X;(c) The median and interquartile range of X.
A random variable X has the linear distribution given bywhere a and b are constants. Show that(a) a = √2b (b) The median of X is (√2 – 1)/√b fx(x) = (a - bx (0
You arrive at a railway station knowing only that trains leave for your destination at intervals of one hour. Find the mean and standard deviation of your waiting time.
The distribution of the number X of independent attempts needed to achieve the first success when the probability of success is 0.2 at each attempt is given byFind the mean, the median and the standard deviation for this distribution. k-1 P(X= k) = (0.2)(0.8) (k = 1, 2, 3, ...)
Find the average sentence length for the sentences with lengths given in Question 1 in Exercises 13.2.6.Data from Question 1A sample of 52 spoken sentences have the following lengths in words:Draw a histogram of the lengths from 1 to 12 words. What do you notice about this histogram? 7, 3, 8, 6,
The distribution of the daily number of malfunctions of a certain computer is given by the following table:Find the mean, the median and the standard deviation of this distribution. Number of malfunctions 0 1 2 3 4 5 6 Probability 0.17 0.29 0.27 0.16 0.07 0.03 0.01
Suppose that the probability distribution for the number of days required to ship a package from London to New York is as follows:Find the mean of this distribution, and the probability that a particular package arrives in less than five days. Number of days Probability 2 3 4 5 6 0.05 0.20 0.35
The wave amplitude X on the sea surface often has the following (Rayleigh) distribution:where a is a positive constant. Find the distribution function and hence the probability that a wave amplitude will exceed 5.5 m when a = 6. - (2) X -exp fx(x) = a 2a 0 (x > 0) (otherwise)
The time interval (X) between successive earthquakes of a certain magnitude has an exponential distribution with density function given bywhere x is measured in days. Find the probability that such an interval will not exceed 30 days. fx(x) = 90 --x/0 0 if x > 0 if x < 0
Suppose that a coin is tossed three times and that the random variable W represents the number of heads minus the number of tails.(a) 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.(b) Find the probability distribution of W,
If the probability density function of a random variable X is given bywhere c is a constant, find(a) The value of c;(b) The distribution function;(c) P(X > 1). fx(x) = [c/x (0
A difficult assembly process must be undertaken, and the probability of success at each attempt is 0.2. The distribution of the number of independent attempts needed to achieve success is given by the product rule asPlot the distribution function and find the probabilities that the number of
At the 18th hole of a golf course the probability that a golfer will score a par four is 0.55, the probability of one under is 0.17, of two under is 0.03, of one over is 0.2 and of two over is 0.05. Plot the (cumulative) distribution function.
Find the distribution of the sum of the numbers when a pair of dice is tossed.
On an infinite chess board with each side of a square equal to d, a coin of diameter 2r < d is thrown at random. Find the probabilities that(a) The coin falls entirely in the interior of one of the squares;(b) The coin intersects no more than one side of a square.
An advertising agency notes that approximately one in fifty potential buyers of a product sees a given magazine advertisement and one in five sees the corresponding advertisement on television. One in a hundred sees both. One in three of those who have seen the advertisement purchase the product,
A system can fail (event C) because of two possible causes (events A and B). The probabilities of A, B and A ∩ B are known, together with the probabilities of failure given A, given B and given A ∩ B. Express the following in terms of these known quantities: (a) P(AUB) (c) P(C| AUB) (b) P(CAN
Part of an electrical circuit consists of three elements K, L and M in series. Probabilities of failure for elements K and M during operating time t are 0.1 and 0.2 respectively. Element L itself consists of three sub-elements L1, L2 and L3 in parallel, with failure probabilities 0.4, 0.7 and 0.5
Three people work independently at deciphering a message in code. The probabilities that they will decipher it are 1/5, 1/4 , and 1/3. What is the probability that the message will be deciphered?
If P(A) = 0.3, P(B) = 0.4 and P(B|A) = 0.5, find (a) P(AB) (b) P(AUB) (c) P(BA)
During the repair of a large number of car engines it was found that part number 100 was changed in 36% and part number 101 in 42% of cases, and that both parts were changed in 30% of cases. Is the replacement of part 100 connected with that of part 101? Find the probability that in repairing an
Two fair coins are tossed once. Find the conditional probability that both coins show heads, given that(a) The first coin shows a head;(b) At least one coin shows a head.
The ‘odds’ in favour of an event A are quoted as ‘a to b’ if and only if P(A) = a/(a + b). The ‘odds against’ are then ‘b to a’ (which is the usual way to quote odds in betting situations).(a) If an insurance company quotes odds of 3 to 1 in favour of an individual 70 years of age
Suppose that you roll a pair of ordinary dice repeatedly until you get either a total of seven or a total of ten. What is the probability that the total then is seven?
In a single throw of two dice, what is the probability of getting(a) A total of 5,(b) A total of at most 5,(c) A total of at least 5?
If a card is drawn from a well-shuffled pack of fifty-two playing cards, what is the probability of drawing(a) A red king(b) A 3, 4, 5 or 6(c) A black card(d) A red ace or a black queen?
The personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a pay rise last year, 248 got increased pension benefits, 173 got both and 43 got neither. Explain why this claim should be questioned.
Two ordinary six-faced dice are tossed. Write down the sample space of all possible combinations of values. What is the probability that the two values are the same? What is the probability that they differ by at most one?
From a pack of fifty-two cards a card is withdrawn at random and not replaced. A second card is then drawn. What is the probability that the first card is an ace and the second card a king?
If A and B are mutually exclusive events and P(A) = 0.2 and P(B) = 0.5, find (a) P(AUB) (b) P(A) (c) P(ANB)
Let the sample space S and three events be defined as S = {car, bus, train, bicycle, motorcycle, boat, aeroplane), A = {bus, train, aeroplane}, B = {train, car, boat}, C = {bicycle}. List the elements of the sets corresponding to the following events: (a) A (b) An Bn C (c) (AUB) n (ANC)
If S is the set {bolt, nut, washer, screw, bracket, flange}, and A and B are sets {bracket, nut, flange} and {bolt, bracket} respectively, then what combinations of A and B produce the following sets as outcomes?(a) {bracket}(b) {flange, bracket, bolt, nut}(c) {washer, bolt, screw}(d) {screw,
Using the data in Figure 13.4:(a) Draw two histograms of temperatures for engine A, first with class boundaries at even numbers, then with boundaries at multiples of five;(b) Draw a cumulative percentage plot for the running time data for engine A and compare it with a similar plot for engine
Figures for a well’s daily production of oil in barrels are as follows:Construct a stem-and-leaf plot with stem labels 19*, 20*, . . . ., 24*. 214, 203, 226, 198, 243, 225, 207, 203, 208, 200, 217, 202, 208, 212, 205, 220
Construct stem-and-leaf plots for the data in Question 2: (a) Using * as a placeholder for the second digit, (b) Using * as a placeholder for 0, 1, 2, 3 and 4 in the second digit and + as a placeholder for 5, 6, 7, 8 and 9.
The following data consists of percentage marks achieved by students sitting an examination:Draw histograms with Class boundaries at intervals of five, 47, 51, 75, 58, 70, 73, 63, 60, 60, 54, 60, 67, 50, 60, 74, 69, 51, 67, 49, 66, 61, 46, 66, 57, 55, 60, 62, 36, 52, 67, 62, 51, 62, 62, 59, 52, 75,
A sample of 52 spoken sentences have the following lengths in words:Draw a histogram of the lengths from 1 to 12 words. What do you notice about this histogram? 7, 3, 8, 6, 10, 6, 2, 9, 5, 8, 2, 7, 1, 8, 5, 4, 12, 9, 3, 6, 2, 8, 2, 10, 7, 4, 11, 9, 8, 2, 6, 1, 3, 11, 7, 8, 1, 4, 2, 9, 7, 3, 8, 5,
Solve, using Laplace transforms, the following differential equations: (a) dx dt d.x + 4+ 5x = 8 cost dt subject to x = (b) sd dt - dx 3- dt d.x dt = 0 at t = 0 - 2x = 6 subject to x = 1 and d.x dt = = 1 at t = 0
(a) Find the inverse Laplace transform of(b) A voltage source Ve–tsin t is applied across a series LCR circuit with L = 1, R = 3 and C = 1/2.Show that the current i(t) in the circuit satisfies the differential equationFind the current i(t) in the circuit at time t ≥ 0 if i(t) satisfies the
Use Laplace transform methods to solve the simultaneous differential equations dx dt dy dt - 5dy = t dt x+ 5 dy - 4y - 2dx = -2 dt subject to x=y= dx dt dy dt = 0 at t = 0.
Solve the differential equationsubject to the initial conditions x = x0 and dx/dt = x1 at t = 0. Identify the steady state and transient solutions. Find the amplitude and phase shift of the steady state solution. dx dr dx + 2 + 2x = cost dt
Resistors of 5 and 20Ω are connected to the primary and secondary coils of a transformer with inductances as shown in Figure 11.19. At time t = 0, with no current flowing, a voltage E = 100V is applied to the primary circuit. Show that subsequently the current in the secondary circuit isFigure
(a) Find the Laplace transforms of(b) Using Laplace transform methods, solve the differential equationgiven that x = 2 and dx/dt = 1 when t = 0. (i) cos(wt + p) (ii) esin(wt + )
(a) Find the inverse Laplace transform of(b) Solve using Laplace transforms the differential equationgiven that y = –3 when t = 0. S-4 s + 4s + 13
Using Laplace transforms, solve the simultaneous differential equationswhere x = 1 and y = 0 when t = 0. d.x dt dy dt + 5x + 3y = 5 sint - 2 cost + 3y + 5x = 6 sint - 3 cost
The charge q on a capacitor in an inductive circuit is given by the differential equationand it is also known that both q and dq/dt are zero when t = 0. Use the Laplace transform method to find q. What is the phase difference between the steady state component of the current dq/dt and the applied
Use Laplace transforms to find the value of x given thatand that x = 2 and dx/dt = –2 when t = 0. d.x 4 + 6x + y = 2 sin 2t dt dx dr + x dy dt = = 3e-21
(a) Use Laplace transforms to solve the differential equationgiven that θ = 0 and dθ/dt = 0 when t = 0.(b) Using Laplace transforms, solve the simultaneous differential equationsgiven that i1 = 1, i2 = 0 when t = 0. d0 +8 dr dt + 160 = sin 2t
The terminals of a generator producing a voltage V are connected through a wire of resistance R and a coil of inductance L (and negligible resistance). A capacitor of capacitance C is connected in parallel with the resistance R, as shown in Figure 11.20. Show that the current i flowing through the
Show that the currents in the coupled circuits of Figure 11.21 are determined by the simultaneous differential equationsFind i1 in terms of t, L, E and R, given that i1 = 0 and di1/dt = E/L at t = 0, and show thatfor large t. What does i2 tend to for large t?Figure 11.21 di L + R(i, - i) + Ri = E
A system consists of two unit masses lying in a straight line on a smooth surface and connected together to two fixed points by three springs. When a sinusoidal force is applied to the system, the displacements x1(t) and x2(t) of the respective masses from their equilibrium positions satisfy the
(a) Obtain the inverse Laplace transforms of(b) Use Laplace transforms to solve the differential equationgiven that y = 4 and dy/dt = 2, when t = 0. (i) s+ 4 s + 2s + 10 (ii) s-3 (s 1)(S-2) -
(a) Determine the inverse Laplace transform of(b) The equation of motion of the moving coil of a galvanometer when a current i is passed through it is of the formwhere θ is the angle of deflection from the ‘no-current’ position and n and K are positive constants. Given that i is a constant and
Two cylindrical water tanks are connected as shown in Figure 11.22. Initially there are 250 litres in the top tank and 50 litres in the bottom tank. At time t = 0 the valve between the two tanks and the valve at the bottom of the lower tank are opened. The flowrate through each of these valves is
In order to transport sensitive equipment a crate is installed inside a truck on damped springs, as shown in Figure 11.23. The suspension system of the truck, including the tyres, may be modelled as a damped spring. The various spring and damper constants are indicated in the figure. The masses
Use the definition of the Laplace transform to obtain the transforms of f(t) when f(t) is given bystating the region of convergence in each case. (a) cosh 2t (b) t (c) 3+1 (d) te
What are the abscissae of convergence for the following functions? (a) er (c) sin 21 (e) cosh 21 (g) e + 1 -51 (i) 3e - 2e-2 + sin 21 (b) e -31 (d) sinh 31 (f) (h) 3 cos 21 - 1 (j) sinh 3r+sin 31
Using the results shown in Figure 11.5, obtain the Laplace transforms of the following functions, stating the region of convergence:Figure 11.5 (a) 5-3t (c) 32t + 4 cos 2t (e)sinh2t (g) 4te-2 (i) te (k) 2 cos 3t + 5 sin 3t (m) sin 31 (0) te " + e 'cos 21 + 3 -41 (b) 7t- 2 sin 3t (d) cosh 3t (f)
Find when F(s) is given by L-'{F(s)}
Using Laplace transform methods, solve for t ≥ 0 the following differential equations, subject to the specified initial conditions. (a) (c) d.x + 3x = e2 subject to x = 2 at t = 0 dt (b) 3- d.x (e) dt dx dr - 4x = sin 2r subject to x = at t = 0 +2^^+5r=1 2dx dr subject to x = 0 and (d) +2 - + y =
Using Laplace transform methods, solve for t ≥ 0 the following simultaneous differential equations subject to the given initial conditions: (a) 2 dx dr (b) (c) 2dx + 4y + 4x-37y=0 2- dr dt d.x dr subject to x = 0 and y = at t = 0 dy + 2 + x - y = 5 sinf dt d.x - 2- dy 2- - 9y=e-2 dr 3dy dr dt
Use the Laplace transform technique to find the transforms I1(s) and I2(s) of the respective currents flowing in the circuit of Figure 11.14, where i1(t)is that through the capacitor and i2(t) that through the resistance. Hence, determine i2(t). (Initially, i1(0) = i2(0) = q1(0) = 0.) Sketch i2(t)
At time t = 0, with no currents flowing, a voltage v(t) = 10 sin t is applied to the primary circuit of a transformer that has a mutual inductance of 1 H, as shown in Figure 11.15. Denoting the current flowing at time t in the secondary circuit by i2(t), show thatand deduce that L{i(t)} = 10s (s +
In the circuit of Figure 11.16 there is no energy stored (that is, there is no charge on the capacitors and no current flowing in the inductances) prior to the closure of the switch at time t = 0. Determine i1(t) for t > 0 for a constant applied voltage E0 = 10 V. t=0 Eo Figure 11.16 1 i (t)
Determine the displacements of the masses M1 and M2 in Figure 11.13 at time t > 0 whenWhat are the natural frequencies of the system? M = M = 1 K = 1, K=3 and K = 9
When testing the landing-gear unit of a space vehicle, drop tests are carried out. Figure 11.17 is a schematic model of the unit at the instant when it first touches the ground. At this instant the spring is fully extended and the velocity of the mass is √(2gh), where h is the height from which
Consider the mass–spring–damper system of Figure 11.18, which may be subject to two input forces u1(t) and u2(t). Show that the displacements x1(t) and x2(t) of the two masses are given by x (1) K B eeeee M x (t) = L-, where x (t) = L-1 u, (t) u(t) x(1) Figure 11.18 Mechanical system of
In each of the following a periodic function f(t) of period 2 is specified over one period. In each case sketch a graph of the function for –4π ≤ t ≤ 4π and obtain a Fourier series representation of the function. (a) f(t) = (b) f(t) = (c) f(t) = T (-
Obtain the Fourier series expansion of the periodic function f(t) of period 2π defined over the period 0 ≤ t ≤ 2π byUse the Fourier series to show that f(t) = (n t) (0 t 2)
The charge q(t) on the plates of a capacitor at time t is as shown in Figure 12.12. Express q(t) as a Fourier series expansion. -2T q(t) Q 77 2T Figure 12.12 Plot of the charge q(t) 3TT 4T
The clipped response of a half-wave rectifier is the periodic function f(t) of period 2 defined over the period 0 ≤ t ≤ 2π byExpress f(t) as a Fourier series expansion. f(t) = [5 sint (0 t ) 0 ( 1 2)
Show that the Fourier series representing the periodic function f(t), where f(t) = (- $is f(t) = {x^ + = - h=1 (a) 2 2 cosnt + sin(2n 1)t (2 1)3 Use this result to show that = (6) M=l n=l$
A periodic function f(t) of period 2π is defined within the domain 0 ≤ t ≤ bySketch a graph of f(t) for –2π (a) f(t) is an even function;(b) f(t) is an odd function.Find the Fourier series expansion that represents the even function for all values of t, and use it to show that f(t) = (0 t
A periodic function f(t) of period 2π is defined within the period 0 ≤ t ≤ 2 by.Draw a graph of the function for –4π ≤ t ≤ 4π and obtain its Fourier series expansion. By replacing t by t – 1/2π in your answer, show that the periodic functionis represented by a sine series of odd
Find a Fourier series expansion of the periodic function f(t) = (-1 < t
A periodic function f(t) of period 2l is defined over one period byDetermine its Fourier series expansion and illustrate graphically for –3l f(t) = K -(1+t) (-1
A periodic function of period 10 is defined within the period –5 Determine its Fourier series expansion and illustrate graphically for –12 f(t) = [0 (-5
Passing a sinusoidal voltage Asinvt through a half-wave rectifier produces the clipped sine wave shown in Figure 12.15. Determine a Fourier series expansion of the rectified wave. - - f(t) A Figure 12.15 Rectified sine wave 3
Obtain a Fourier series expansion of the periodic functionand illustrate graphically for –3T f(t)=t (-T
Determine a Fourier series representation of the periodic voltage e(t) shown in Figure 12.16. e(t). T E -2T Figure 12.16 Voltage e(t) T 2T 3T
Show that the half-range Fourier sine series expansion of the function f(t) = 1, valid for 0 Sketch the graphs of both f(t) and the periodic function represented by the series expansion for –3π 4 f = n=1 sin(2n - 1)t 2n - 1 (0
Determine the half-range cosine series expansion of the function f(t) = 2t – 1, valid for 0 < t < 1. Sketch the graphs of both f (t) and the periodic function represented by the series expansion for –2 < t < 2.
The function f(t) = 1 – t2 is to be represented by a Fourier series expansion over the finite interval 0 < t < 1. Obtain a suitable(a) Full-range series expansion;(b) Half-range sine series expansion;(c) Half-range cosine series expansion.Draw graphs of f(t) and of the periodic functions
A function f(t) is defined byand is to be represented by either a half-range Fourier sine series or a half-range Fourier cosine series. Find both of these series and sketch the graphs of the functions represented by them for –2π f(t) = t - t (0 t n)

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