In this brief Article, using the elementary theory of differential equations as well as some basic economic theory, we will develop several estimates for national health expenditures for the United States: one using a linear model and three using non-linear models. We will derive the nonlinear models first and then compare them to the linear one in order to see if they differ significantly. While these estimates are for the United States, the methods used here, because they are robust, could be used for any country. Statistical information may be obtained from the World Bank databases which store health statistics by country [1].

What we will do here is estimate the total health costs as a percentage of gross domestic product (GDP) if no further copayments are required. In other words, we are seeking to estimate the total cost of health care as a percentage of GDP when all health care costs are covered by insurance and government subsidy. Several models will be discussed here since such estimates may be made using a variety of assumptions. There is no ‘best’ model, although such a decision is possible when comparing the estimates to actual data.

Recently a friend kindly made me a birthday card whose background consisted of rows and rows of digits, some 3000 in all. There appeared to be no discernible pattern in the digits. Perhaps they had been taken from a table of random numbers. They were certainly not the opening digits of the decimal parts of π or , although they might, so far as I knew, have been consecutive digits of either number in some section remote from the decimal point.

On thinking about this, I realised that they must be the opening digits of the decimal part of the square root of some whole number. Indeed, they must be the opening digits of the decimal parts of the square roots of infinitely many positive integers. It is remarkably easy to prove this and the argument is simple enough to be appreciated and understood in school classrooms at GCSE level.

If no two of three non-collinear points share the same x-coordinate, then the parabola y = a2x2 + a1x + a0 through the points is easily found by solving a system of linear equations. That is but one of an infinite number of parabolas through the three points. How does one find the other parabolas? In this note, we find all parabolas through any three non-collinear points by reducing the problem to finding the equation of a parabola by using rotations.

The parabola y = a2x2 + a1x + a0 has an axis of symmetry parallel to the y-axis. Other parabolas have an axis of symmetry that is parallel to some line y = mx. We focus on the angle θ that the axis of symmetry makes with the y-axis, as in Figure 1, so that tanθ = 1/m. To find the parabola associated with θ that goes through three non-collinear points, we rotate the three points counterclockwise by θ, find the equation of the parabola, and then rotate the parabola (and the three points) counterclockwise back by −θ so that the parabola goes through the original points.

Finding an expression for the length of a curve is one of the simpler geometric applications of the integral. If f is a function with a continuous derivative, then the expression gives the length of the curve y = f (x) on an interval [a, b]. However, after writing out the integrand for familiar functions such as y = x2 and y = sin x, it quickly becomes apparent that, in general, finding an antiderivative is a challenge. Of course, a computer can give accurate approximations for the value of the integral for the length of a curve, but it would be nice to find the exact length rather than a decimal approximation. In his work on geometry (from 1637), Descartes stated that he believed it was not possible to determine the exact lengths of curves. However, just twenty years later, William Neile was able to find the length of arcs of semicubical parabolas (see Katz [1]). These curves have the form y = kx3/2 and are usually the first examples or exercises given to students since the resulting integral is very easy to compute. In this paper, we are going to examine this curve and other related curves and consider problems such as the following: find rational numbers a and b so that the length of the curve over the interval is an [a, b] integer. As we shall see, problems such as this provide a variety of opportunities for undergraduate students to explore some interesting mathematics arising from a few simple and accessible questions.

Some forty years ago my wife Christine and I considered the problems in this article, which involves a fair amount of computation. Computing facilities were not good then, so we considered instead the problems in [1] in which we showed, without using computers, that there were 64703 ways to make up £1 using coins; this was before the introduction of the 20p and £1 coins, and the ½p coin was in circulation. If Christine were still with us, this would have been another piece of joint work. I therefore dedicate this article to her memory.

The design of a coinage system depends on considerations we give to various criteria: for example, the number of denominations for the coins, the maximum number of coins required to deliver any given amount in a range, or the required number of coins averaged over the range; see also §3.

In a recent and illuminating article that provided much food for thought [1], the problem of tethering a goat at the edge of a circular pasture so as to restrict its attentions to only one half of the grazing supply was elegantly addressed and developed further to embrace the corresponding three-dimensional scenario involving a bird. The exercises resulted in mathematical formulations that required the use of numerical methods to extract practical results. Following the article, various questions and different scenarios sprang to my mind. The following poser perhaps best illustrates one of these, and it is the purpose of this Article to address this particular conundrum:

It used to be the case in some jurisdictions that a rhythmic knell of the Angelus bell would mark the onset of dawn, noon and the passing of a day at dusk. Between these times there is daylight whose duration varies from place to place and from day to day and which can be predicted either exactly or approximately. An illuminating problem concerning the number of daylight hours at a winter solstice in London was posed recently and answered in the Problem Corner of The Mathematical Gazette [1]. It was shown, for example, that a calculation of daylight hours rested strictly upon the numerical solution to a transcendental trigonometric equation. Related references to earlier works in the Gazette involving a point source Sun were also given.

The purpose of this note is multifold. First, it is to point out that the above-cited equation might be viewed also as a “Sun-ray-to-Earth tangency condition”. Such a condition was developed earlier by the author in a publication that is now defunct [2], and so for completeness the steps necessary to establish the condition will be produced again here. Second, it will be evident from the manner of its derivation that the governing equation is valid at all orbit points, not just at a given solstice.

This Article is inspired by a problem that appeared recently in the American Mathematical Monthly, namely 11972 in [1, p. 369]. The problem asks the readers to prove that if r is the inradius of a tetrahedron, and if r1, r2, r3, r4 are its exradii, then

By taking r = r1 = r2 = r3 = r4 = 1, one sees that (1) is not true for all positive numbers r, r1, r2, r3, r4. This is not surprising, since r is dependent on r1, r2, r3, r4 by the elegant relation

which we shall prove in Theorem 2 below; see also, for example, [2, (5), §266, p. 92], [3. §41, 2°, (1), p. 76] and [4, Problem 6′(i), p. 39]. Using this relation, one can rewrite (1) as

Intuitively, the four numbers r1, r2, r3, r4 are independent, and one may thus ask whether the inequality (3) holds for all positive numbers r1, r2, r3, r4 regardless of being the exradii of some tetrahedron or not.

The trapezium (trapezoid in American English) is one of the seven most well-known quadrilaterals that is studied in school geometry. However, it is always assumed that it is convex. In this note we shall derive metric formulas for the most important quantities in crossed trapezia and compare them to the similar formulas for convex trapezia. The corresponding investigation regarding crossed cyclic quadrilaterals was conducted in [1]. Any crossed quadrilateral has a pair of opposite sides intersecting and both of its diagonals are outside of itself.

In this survey we give very simple proofs of Kepler's Laws and other facts about central force fields using only Newton's second law, Newton's law of universal gravitation, basic notions of vector calculus, and an elementary double integral.

Hopefully, this article will help undergraduate students of mathematics and engineering who wish to understand these fundamental scientific discoveries.

In many textbooks (see, for instance, [1, 2, 3, 4, 5]), Kepler's Laws are obtained using conservation of energy and angular momentum, differential equations, mobile reference systems, or notions not so well-defined such as differentials or ‘infinitesimal elements’. Some of the arguments appear to be rather involved if one is not accustomed to them, whereas the proof of Kepler's Laws may actually be obtained from quite simple facts.

The circles C1, & , Cn form a chain of length n if Ci touches Ci + 1, for i = 1, & , n − 1, and the chain is closed if also Cn touches C1. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain. If Ci touches S at Pi, then P1, & , Pn are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if Pi = Pj, then the line PiPj should be interpreted as the tangent S to at Pi.

The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P1P4, P2P5, P3P6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.

When studying geometry, it is natural to choose an object and try to determine what properties it has. Take for example a trapezium (trapezoid in American English): a quadrilateral with at least one pair of opposite parallel sides. Then we probably want to know how to calculate its area, the length of the diagonals, what other properties the diagonals might have, if it has some angle properties, how to calculate the length of the midsegment and so on. All these are interesting investigations. But when they are done (and in many textbooks the story is not allowed to continue even this far), then it is very common to consider the exploration of the trapezium finished. The truth is that here we are only half way. The most interesting part of the journey is the second half. So far we only know necessary conditions in the trapezium, that is, given that we have a quadrilateral that is a trapezium, what properties it has. But what about the reverse question? Which of these properties are unique to the trapezium, that is, which are the sufficient conditions? Or in other words: which properties characterise trapezia and separate them from all the other quadrilaterals? To me that is the really interesting question!