Evaluate:
The centres of two large solid hemispherical radar domes of radii \(a\) and \(b\) are at a distance \(c\) apart. An aesthete wishes to stand at the point, on the line of centres between the two hemispheres, at which the least amount of hemispherical surface area is visible. Where should he stand?
Find the surface area of each of the two spheroids that are obtained by rotating the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1, \quad (b < a),\] about its major and minor axes. Express the areas in terms of \(a\) and the eccentricity \(e\) of the ellipse. In each case verify that the limit of the area, as \(e \to 0\), is \(4\pi a^2\).
The following is a simple theory for the decompression of divers: When the diver is at a depth \(b\), the pressure \(A\) of gas in his lungs is \((1+b/10)\), and the pressure \(P\) of gas dissolved in his body tissues is governed by the equation \(\frac{dP}{dt} = k(A-P)\), where \(k\) is a positive constant. The risk of 'bends' is proportional to \(P/A\) and ceptable if \(P/A < 2\). The diver is at a depth \(D\), with \(P = A = (1+D/10)\), and wishes to ascend to the surface at a constant speed \(s\). Show that the risk is acceptable provided \[s(1 - e^{-kD/s}) < 10k.\]
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.