Mathematical Methods for Physics and Engineering-Solution Manual 3rd. Edition by K. F. Riley, M. P. Hobson | 9780521679732 | 0521679737 | Holooly

Mathematical Methods for Physics and Engineering-Solution Manual [EXP-676]

383 SOLVED PROBLEMS

A and B each have two unbiased four-faced dice, the four faces being numbered 1, 2, 3 and 4. Without looking, B tries to guess the sum x of the numbers on the bottom faces of A’s two dice after they have been thrown onto a table. If the guess is correct B receives x^2 euros, but if not he loses x

Verified Answer:

We first calculate the probabilities p(x) and the ...

Question: 5.19

A barn is to be constructed with a uniform cross-sectional area A throughout its length. The cross-section is to be a rectangle of wall height h (fixed) and width w, surmounted by an isosceles triangular roof that makes an angle θ with the horizontal. The cost of construction is α per unit height

Verified Answer:

The cost always includes 2αh for the vertical wall...

Question: 26.19

A block of wood contains a number of thin soft-iron nails (of constant permeability). A unit magnetic field directed eastwards induces a magnetic moment in the block having components (3, 1,−2), and similar fields directed northwards and vertically upwards induce moments (1, 3,−2) and (−2,−2, 2)

Verified Answer:

The magnetic moment M, the permeability µ and the ...

A boy is selected at random from amongst the children belonging to families with n children. It is known that he has at least two sisters. Show that the probability that he has k − 1 brothers is \frac { (n-1)! }{ ({ 2 }^{ n-1 }-n)(k-1)!(n-k)! }, for 1 ≤ k ≤ n − 2 and zero for other values of k. Assume that boys and girls are equally likely.

Verified Answer:

The boy has n − 1 siblings. Let

{A}_{j}[/la...

A cast iron pipe has an inside diameter of d= 208mm and an outside diameter of D= 236mm. The length of the pipe is L= 3.0m. The coefficient of thermal expansion for cast iron is α1 = 12.1 * 10^-6/°C. Determine the dimension changes caused by an increase in temperature of 70°C.

Verified Answer:

The thermal strain caused by a temperature increas...

Question: 6.9

A certain torus has a circular vertical cross-section of radius a centred on a horizontal circle of radius c (> a). (a) Find the volume V and surface area A of the torus, and show that they can be written as V = π²/4 (r²o − r²i )(ro − ri), A = π²(r²o − r²i ), where ro and ri are, respectively, the

Verified Answer:

(a) The inner and outer radii of the torus are [la...

A circular disc of radius a is heated in such a way that its perimeter ρ = a has a steady temperature distribution A+Bcos^2ϕ, where ρ and ϕ are plane polar coordinates and A and B are constants. Find the temperature T(ρ,ϕ) everywhere in the region ρ < a.

Verified Answer:

This is a steady state problem, for which the (hea...

Question: 3.27

A closed barrel has as its curved surface the surface obtained by rotating about the x-axis the part of the curve y = a[ 2 − cosh(x/a) ] lying in the range −b ≤ x ≤ b, where b<a cosh−1 2. Show that the total surface area, A, of the barrel is given by A = πa[ 9a − 8a exp(−b/a) + a exp(−2b/a) − 2b ].

Verified Answer:

If s is the length of the curve defining the surfa...

A continuous random variable X has a probability density function f(x); the corresponding cumulative probability function is F(x). Show that the random variable Y = F(X) is uniformly distributed between 0 and 1.

Verified Answer:

We first note that, as F(x) is a cumulative probab...

Question: 22.13

A dam of capacity V (less than πb²h/2) is to be constructed on level ground next to a long straight wall which runs from (−b, 0) to (b, 0). This is to be achieved by joining the ends of a new wall, of height h, to those of the existing wall. Show that, in order to minimise the length L of new wall

Verified Answer:

The objective is to chose the wall profile, y = y(...

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