The barrel of a gun may be considered as a tube of length \(L\), closed at one end, and of uniform circular cross section of area \(A\). The rear surface of the bullet is at a distance \(x\) from the closed end, and \(x = x_0\) when the gun is fired. The pressure in the gun is \(P\), and \(P = P_0\) immediately after firing. Subsequently \(P\) obeys the equation \[PV^{\gamma} = \text{constant},\] where \(V = Ax\) is the volume of propellant gas, and \(\gamma\) is a constant \(\geq 1\). The equation of motion of the bullet is \[m\frac{d^2x}{dt^2} = AP.\] Find the velocity of the bullet when it leaves the barrel, for all values of \(\gamma \geq 1\).
Evaluate \[\int_0^1 \frac{u^{\frac{1}{2}}}{(1+u)^{\frac{1}{2}}}\,du.\]
Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a square. This result changes if, instead of maximizing the sum of lengths of sides of the rectangle, we seek to maximize the sum of \(n\)th powers of the lengths of those sides, for an integer \(n > 1\). What happens? Justify your answer.
By considering the derivative of \(x - \sin x\) show that \(x \geq \sin x\) for all \(x \geq 0\). By considering the repeated derivatives of \(\sin x - x + x^3/3!\) show that \(\sin x \geq x - x^3/3!\) for all \(x \geq 0\). More generally, show that \[\sum_{r=0}^{2m} (-1)^r \frac{x^{2r+1}}{(2r+1)!} \geq \sin x \geq \sum_{r=0}^{2m-1} (-1)^r \frac{x^{2r+1}}{(2r+1)!}\] for all \(x \geq 0\) and \(m \geq 1\). Deduce that \[\left|\sum_{r=0}^{n-1} (-1)^r \frac{x^{2r+1}}{(2r+1)!} - \sin x\right| \leq \frac{x^{2n+1}}{(2n+1)!}\] for all \(x \geq 0\). [The power series expansion of \(\sin x\) must not be used.]
The president of the republic must have a son and heir. It may be assumed that each baby born to him is equally likely to be a boy or a girl, irrespective of the sexes of his previous children. Let \(\mu\) and \(\sigma^2\) denote the mean and variance, respectively, of the number of children in his family, if he decides to have no more children once he has a son. Evaluate \(\mu\). Now suppose that he decides to have no more children once he has exactly \(r\) sons. Express the mean and variance of \(C\), the number of children in his family, in terms of \(\mu\) and \(\sigma^2\). By considering the numbers of boys and girls among \(2r - 1\) children, or otherwise, show that \(\Pr[C < 2r] = \frac{1}{2}\).
Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed random variables with mean \(\beta\), taking integer values in the range \(1, 2, \ldots, K\). For each \(m\), \(1 \leq m \leq n\), let \(S_m = X_1 + X_2 + \cdots + X_m\). Prove that \(E(X_r/S_m) = 1/m\) for \(r = 1, 2, \ldots, m\). Hence show that, if \(m \leq n\), \(E(S_m/S_n) = m/n\) and \(E(S_n/S_m) = 1 + (n-m)\beta E(1/S_m)\).
The average weight in grams of the contents of a sachet of instant mashed potato varies between batches, because of the variable quality of the synthetic feedstock. Within a given batch, the weights of the sachets are independently and normally distributed, with common unknown mean \(m\) and standard deviation \(0 \cdot 1\). In order to check the weight of a given batch, the manufacturer weighs the contents of 25 sachets, obtaining an average weight of \(4 \cdot 92\). Does this give him good grounds for rejecting the hypothesis that \(m\) is really 5? He now decides upon the policy of rejecting a batch whenever the average weight of a sample of \(N\) sachets falls below \(T\). If \(N\) and \(T\) are to be chosen so that the probabilities of wrongly rejecting a batch with \(m = 5\) and of wrongly accepting a batch with \(m = 4 \cdot 95\) are both less than \(0 \cdot 05\), what values would you choose to make \(N\) as small as possible?
A machine produces boiled sweets in large batches. Each batch is either satisfactory, and contains no sub-standard sweets, or defective, when a known proportion \(p\) of the sweets are tasteless. The cost of rejecting a defective batch immediately after production is \(K\); however, if a defective batch is not detected and reaches the customer, it costs \(MK\) to replace it and to recover lost goodwill, where \(M > 1\). The quality control officer decides to test each batch by removing, for tasting, a random number \(N\) of sweets selected from it at random, at a cost of \(c\) per sweet, where \(N\) has a Poisson distribution with mean \(\lambda\). A batch is rejected if any of the \(N\) sweets proves to be tasteless. Show that his chance of detecting a defective batch is \(1 - e^{-\lambda p}\). If the proportion of defective batches produced is known to be \(\alpha\), show that the expected running cost per batch is \(c\lambda + \alpha K[1 + (M-1)e^{-\lambda p}]\). Find the value of \(\lambda\) that minimizes the expected cost (a) if \(c < \alpha p K(M-1)\), and (b) if \(c \geq \alpha p K(M-1)\).
Prove that \[\int_0^{2\pi} \sin nx \sin mx\, dx = 0\] when the positive integers \(n\) and \(m\) are not equal, and evaluate the integral for the case when \(n = m\). Let \(f(x)\) be a periodic function with period \(2\pi\), which may be expressed in the form \[f(x) = \sum_{n=1}^{\infty} a_n \sin nx\] for some constants \(a_n\). Use the results of the first part to obtain expressions for the \(a_n\) in terms of \(f(x)\) by multiplying by \(\sin mx\) and integrating term by term. [You may assume that this procedure is justified.] We now seek, for fixed \(N\), to choose \(a_n\) so that the sum \[\sum_{n=1}^{N} a_n \sin nx\] approximates \(f(x)\) as closely as possible, in the sense that \[\int_0^{2\pi} \left\{f(x) - \sum_{n=1}^{N} a_n \sin nx\right\}^2\, dx\] is minimal. By differentiating with respect to each \(a_n\) separately, show that the solution is given by \(a_n = \hat{a}_n\).
An arthritic squash player cannot move from the point where he is placed initially, and can project the ball only with a fixed velocity in a fixed direction. Since no-one will play with him he bounces the ball back exactly to himself, with a single bounce off a wall, not the floor or ceiling). If the coefficient of restitution at the bounce is \(e\), show that the distance from the wall at which he should have himself positioned is proportional to \(e/(1+e)\).