Problems

A survey is conducted among \(n\) people in order to examine whether there is any association between smoking and lung cancer. The following data are obtained

Here \(n_{11}\) is the number of people who both smoke and have cancer, with \(n_{12}\), \(n_{21}\), \(n_{22}\) defined in the obvious way. It is desired to compress the table \((n_{ij})\) into a single real number \(\delta = \delta(n_{ij})\) which indicates the association between smoking and cancer. Two medical experts are consulted, on the choice of the function \(\delta(n_{ij})\). The first says that \(\delta(n_{ij})\) should be determined uniquely by the pair \((n_{11}/n_{1.}, n_{21}/n_{2.})\), and the second says that \(\delta(n_{ij})\) should be determined uniquely by the pair \((n_{11}/n_{.1}, n_{12}/n_{.2})\). Show that if we choose the function \(\delta(n_{ij})\) to satisfy both the experts, and if \((a_{ij})\), \((b_{ij})\) are two tables for which \[a_{11}a_{22}/a_{12}a_{21} = b_{11}b_{22}/b_{12}b_{21}\] then \(\delta(a_{ij}) = \delta(b_{ij})\). If each of the \(n's\) is fairly large, describe the association between smoking and lung-cancer if

1. \((n_{11}n_{22})/(n_{12}n_{21}) \simeq 1\),
2. \((n_{11}n_{22})/(n_{12}n_{21}) \gg 1\).

Suppose a profit-maximising firm produces a perishable and homogeneous good from which the net revenue per unit sold becomes negative if output exceeds a certain level. If a sales tax of \(t\) pence is imposed on each unit, will the firm produce more or less? Justify your answer.

By vector methods, or otherwise, show that the medians of a triangle are concurrent (at the 'centroid'). Each vertex of a tetrahedron is joined to the centroid of the opposite face. Show that the resulting four lines are concurrent. At each vertex of a tetrahedron a force acts which is towards and perpendicular to the opposite face, and has magnitude proportional to the area of that face. Show that the system of forces is in equilibrium.

Four equal stretched strings \(X_0X_1\), \(X_1X_2\), \(X_2X_3\), \(X_3X_4\), each of natural length \(l\), and modulus of elasticity \(\lambda lm\), lie in a straight line on a smooth horizontal table. The ends \(X_0\), \(X_4\) are fixed, and masses \(m\), \(nm\), \(m\) are attached to the points \(X_1\), \(X_2\), \(X_3\), respectively. The system performs oscillations along the line of the springs. Determine the equations of motion for the masses in terms of their displacements from their equilibrium positions. Show that if all the masses oscillate with the same period \(2\pi/p\), then in order to have a non-trivial solution, either \(p^2 = 2\lambda\), or \(p^2\) satisfies the equation \[(2\lambda - np^2)(2\lambda - p^2) = 2\lambda^2.\]

An axle with perfectly smooth bearings carries a gear-wheel with radius \(a_1\), and the total moment of inertia of the system is \(I_1\). A second similar system is described by parameters \(a_2\) and \(I_2\). The axles are mounted parallel in a rigid piece of machinery, their separation being a little greater than \(a_1 + a_2\), and are rotating with angular velocities \(\omega_1\) and \(\omega_2\). The separation is then reduced, bringing the gear wheels into mesh. Find the loss of energy resulting from the impact. Find also the impulsive couple which must be exerted on the machine as a whole to keep it stationary.

A satellite rotates in a circular orbit around the earth with a period of one day. Find the radius of its orbit. Three such satellites rotate in the earth's equatorial plane. If the satellites lie at the corner of an equilateral triangle, find the largest angle of latitude such that all places on earth at this latitude are visible from at least one of the satellites. [\(g = 9.8\) m/sec\(^2\); radius of the earth \(= 6.4 \times 10^6\) m.]