A hospital buys batches of a certain tablet from a pharmaceutical company. A tablet is considered unsatisfactory if it contains more than 1 microgram of arsenic. It is known that within any batch of tablets the arsenic content is normally distributed with standard deviation 0.05 micrograms about a mean which depends on the batch. From every batch the hospital randomly selects \(n\) tablets for analysis, and rejects the batch if the mean arsenic content of the \(n\) tablets is greater than \(C\). What values should be chosen for \(n\) and \(C\) if the desired chances of rejecting batches with 0.1\% and 1\% of defective tablets, respectively, are 20\% and 90\%?
Tests are to be carried out to discover which of a large number of people have a particular disease. To keep the number of tests low, samples of blood from 40 people are mixed and tested together. If the test indicates that the disease is absent, all 40 people are free from it, but if the test shows that the disease is present, all 40 people are retested individually. Assuming that there is a constant and independent chance \(p\) that a person has the disease, determine the mean number of tests that have to be carried out. The following modified procedure is proposed with the aim of reducing the number of tests: whenever the group test shows that the disease is present, samples from 20 of the group are mixed and tested, and samples from the other 20 are then tested individually. Either or both sets of people are then tested individually if necessary. Show that this procedure does result in a smaller mean number of tests if \(p\) is small enough. Can you suggest any way of improving the procedure further?
Two opponents play a series of games in each of which they have an equal chance of winning. The loser of each game pays the winner one unit of capital. The first player begins with \(k\) units of capital and the second player has all \(\alpha\) units of capital. Let \(p_k\) be the probability that the first player wins the series. Write down a relation between \(p_{k-1}, p_k\) and \(p_{k+1}\); and hence show that \(p_k = k/\alpha\).
A shopkeeper has to meet a continuous demand of \(r\) units per unit of time from his customers. At intervals of \(T\) units of time, he buys a quantity of \(Q\) units from a wholesaler, where \(Q \geq rT\). The cost of placing the order is \(a\) pounds and its cost per unit is \(b\) pounds. If he runs out of stock at any unit time, his customers go elsewhere (at no cost to him per unit of time); but as soon as his shop is set again (through loss of customers, or other business) for the period during which the capital tied up against these losses he makes a net profit on this line of business if \(p^2 > 2ab/r\), where \(p\) is the amount of money per unit sold. Show that he can make a maximum profit per unit time of \(X\) which will maximise his profit per unit time.
Two smooth planes meet at right angles in a horizontal line. A rod, whose density is not necessarily uniform, is placed above this line and perpendicular to it, and rests on the planes. If the steeper plane is inclined at an angle \(\theta\) to the horizontal, find the equilibrium positions of the rod. Discuss explicitly the following special cases:
A rain-drop falls through air containing stationary infinitesimal water droplets. The volume-concentration of droplets is \(c\) (that is, they occupy a fraction \(c\) of the unit of space). The rain-drop maintains its spherical shape during the motion, and coalesces with all the droplets in its path. Show that, at time \(t\), its velocity \(v\) and radius \(r\) satisfy the equations \begin{equation*} \frac{dr}{dt} = \frac{cv}{4}, \quad \frac{d^2}{dt^2}r^4 = cgr^3. \end{equation*} The drop starts from rest as an infinitesimal droplet. Assuming that its acceleration is uniform, show that it is \(g/7\).
By writing \(x = r\cos\theta\) and \(y = r\sin\theta\) (where \(r\), \(\theta\) are polar coordinates at origin \(O\)), or otherwise, show that the components of acceleration of a particle \(P\) along and perpendicular to \(OP\) are \begin{equation*} \ddot{r}-r\dot{\theta}^2 \quad \text{and} \quad r\ddot{\theta}+2\dot{r}\dot{\theta} \end{equation*} respectively, where dots denote differentiation with respect to time. A particle of unit mass is moving under the action of a force \(F(1/r)\) directed toward the origin. Show that \begin{equation*} r^2\dot{\theta} = h, \end{equation*} where \(h\) is a constant, and also that, if \(u = 1/r\), \begin{equation*} h^2u^2\left(u+\frac{d^2u}{d\theta^2}\right) = F(u). \end{equation*} Find the equation of the path of the particle if \(F(u) = Au^3\), where \(A < h^2\).
An aircraft is flying above a plane inclined at an angle \(\alpha\) to the horizontal. A smooth sphere is dropped from the aircraft when it is travelling horizontally with speed \(u\) and at such a height as to make the sphere impinge normally on the plane. Show that the sphere travels a distance \begin{equation*} \frac{2u^2e^2}{g\sin\alpha\cos^2\alpha(1-e)^2} \end{equation*} along the plane before it ceases to bounce. (Here \(e\) is the coefficient of restitution between the sphere and the plane.)
State the laws of conservation of linear momentum and energy for the motion and collision of perfectly elastic smooth spherical particles referred to a fixed frame of reference \(F\) in the absence of external forces. A second frame of reference \(F'\) moves with velocity \(\mathbf{v}+\mathbf{a}t\) relative to \(F\) where \(\mathbf{v}\) and \(\mathbf{a}\) are constant vectors and \(t\) is time. Prove that the conservation laws holding in \(F\) hold also in \(F'\) if and only if \(\mathbf{a} = \mathbf{0}\). What alternative equations of motion hold in \(F'\) if \(\mathbf{a} \neq \mathbf{0}\)?
Three linear springs each of modulus \(\lambda\) and natural length \(l\) are connected end to end and lie in a straight line on a smooth horizontal table. At each of the two points where the springs join, a mass \(m\) which is free to move is attached. The two ends of the composite spring are attached to the table, so that in equilibrium the springs are all stretched. If \(x\) and \(y\) denote small displacements of the masses from their equilibrium positions along the line of the springs, show that \begin{equation*} ml(\ddot{x}+\ddot{y})+\lambda(x+y) = 0 \end{equation*} and \begin{equation*} ml(\ddot{x}-\ddot{y})+3\lambda(x-y) = 0. \end{equation*} Describe exactly the subsequent motion if, at \(t = 0\), one of the masses is given a sudden unit velocity towards the second which is itself stationary.