Sketch on the same diagram the curves given in polar co-ordinates \((r, \theta)\) by the equations \(r = \frac{1}{2}a(1 + \cos \theta)\) and \(r = a\theta\) (\(a > 0\), \(0 \leq \theta \leq 2\pi\)). Find the area of the region consisting of all those points \((r,\theta)\) such that \(\frac{1}{3}\pi \leq \theta \leq 2\pi\) and \(\frac{1}{2}a(1 + \cos \theta) \leq r \leq a\theta\).
The section of the curve \(y = \cosh x\) between \(x = 0\) and \(x = a\) is rotated about the \(x\)-axis. Prove that the numerical value of the curved surface area thus obtained is twice that of the volume enclosed. The curve is now rotated about the \(y\)-axis. Calculate the ratio of the numerical values of volume to curved surface area, and show that in this case it depends on \(a\).
In a relay race the baton cannot be passed successfully between two runners unless they are in the same place, travelling at the same speed. The race is run on a straight track and the team manager watches from a point \(O\). Runner \(A\), carrying the baton, passes the manager at time \(t = 0\). \(A\) is running at the speed \(\lambda v_0\), but as he passes the point \(O\) he begins to slow down. His deceleration at any subsequent point is numerically equal to his distance \(s\) from \(O\). As \(A\) passes him, the manager observes \(B\) (to whom the baton is to be passed) at a distance \(s_0\) ahead of \(A\) travelling at a steady speed \(v_0\). Assuming that \(B\) will maintain this speed, prove that the baton can be passed successfully provided \(\lambda \geq 1\) and $$s_0 = v_0\{(\lambda^2-1)^{\frac{1}{2}} - \cos^{-1}(\frac{1}{\lambda})\}.$$ Hence show that if \(\lambda > 1\), a value of \(s_0 > 0\) can be chosen in such a way that a successful hand-over can be made.
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: