Show that \(\iiint dxdydz = 4\pi abc/3\) where the integral is over the space enclosed by the surface \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \quad (a,b,c>0).\] Use this result to calculate \(\iiiint dxdydzdt\) over the space enclosed by the surface \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} + \frac{t^2}{d^2} = 1 \quad (a,b,c,d>0).\]
A proof reader is checking galley-proofs. The number of misprints on a galley is random and has a Poisson distribution with mean \(\mu\). The probability that he detects any one misprint is \(p = 1 - q\), and his result with each misprint is independent of his results with the others. Show that the number of misprints detected (\(X\), say) and the number undetected (\(Y\), say) on a galley are independent random variables with Poisson distributions with means \(p\mu\) and \(q\mu\) respectively.
Mesdames Arnold, Brown, Carr and Davies regularly write gossip letters to each other. When one knows some gossip, she promptly writes about it to a random one of the others whom she does not know already knows it. Since all four are discreet, none ever reveals the source of her information, so it is possible for anyone to re-hear, from one of the others, something she has already passed on; the last letter in a series is written when its recipient then knows that all the others know. One day Mrs Arnold overhears something, and promptly writes off about it. By considering a diagram of the possibilities, answer the following questions:
The two random variables \(U\) and \(V\) are independent and each is uniformly distributed on \((0, 1)\). The random variables \(X\) and \(Y\) are defined by \(X = \log_e(1/U)\), \(Y = \log_e(1/V)\). Prove that the probability that \(X + Y \leq z\) is \[\int_0^z te^{-t}dt \quad (z > 0).\]
An entomologist measures the lengths of 8 specimens of each of two closely related species of bees. His measurements of species \(A\) and of species \(B\) have mean values 15 mm and 17 mm respectively. If he believes that in each species length is normally distributed with standard deviation 2 mm, should he conclude that the mean lengths of the two species differ? What procedure should he use if he does not know the standard deviation (though still believing it to be the same for both species)?
A breakdown truck tows away a car of mass \(m\) by means of an extensible rope whose unstretched length is \(l\) and whose modulus of elasticity is \(\lambda\). Initially the rope is slack and the car stationary; the truck then moves off with speed \(v\) which it maintains constant. The movement of the car is opposed by a constant frictional force \(F\). Determine the motion of the car as a function of time elapsed from the instant the rope becomes taut.
A smooth wedge of mass \(M\) is free to slide on a smooth horizontal plane and has one face inclined at an angle \(\alpha\) to the horizontal. A smooth particle of mass \(m\) is placed on this inclined face of the wedge. The particle and the wedge are initially at rest. Prove that the particle moves in a straight path inclined to the horizontal at an angle \[\tan^{-1}\left[\left(1+\frac{m}{M}\right)\tan\alpha\right].\] Find the velocity of the wedge when the particle has fallen a vertical height \(h\).
A uniform circular disc of mass \(M\) and radius \(a\) is free to rotate about a fixed vertical axis through its centre and perpendicular to it. A shallow groove is cut in the upper surface of the disc along a diameter. Two insects each of mass \(m\) are together on the upper surface of the disc at one end of the groove and the disc is rotating with angular velocity \(\Omega\). At time \(t = 0\) one insect starts to crawl along the groove with uniform velocity \(V\) relative to the disc. Show that when it reaches the other end of the diameter the disc has rotated through an angle \[\frac{2a}{V}\sqrt{\frac{2m}{M+2m}}\left\{\Omega\left(2+\frac{M}{2m}\right)\right\}\tan^{-1}\sqrt{\frac{2m}{M+2m}}.\] If at time \(t = 0\) the other insect starts to crawl round the circumference in the direction of rotation of the disc at such a constant velocity relative to the disc that it arrives at the far end of the diameter at the same instant as its companion, find the angle the disc has turned through when they meet.
Two uniform rough cylinders each with radius \(a\) and mass \(M\) lie touching each other on a rough horizontal table. A third identical cylinder lies on these two. The end faces of all three cylinders are coplanar. The coefficient of friction for all pairs of surfaces in contact has the same value, \(\mu\). Outward horizontal forces \(P\) are applied to the axes of both the lower cylinders. Find the greatest value that \(P\) can have before slipping occurs. Show that when \(\mu = \sqrt{3}\), this value is \(\frac{Mg}{2}\left(1+\frac{1}{\sqrt{3}}\right)\).
A large massive circular cylinder, radius \(a\), rotates about its axis with constant angular velocity \(\Omega\). A projectile is launched from the inside curved surface in a plane perpendicular to the axis of the cylinder with velocity \(V\) and elevation \(\alpha\) relative to the cylinder. Show that the particle hits the cylinder again after a time \begin{equation*} \frac{2aV\sin\alpha}{V^2+a^2\Omega^2+2aV\Omega\cos\alpha}. \end{equation*} Write down the condition that the projectile passes through the axis of the cylinder and find in this case the set of solutions to the condition that the particle strikes the cylinder at its launching point. [You may ignore the effects of gravity.]