1.2.4 Including the results of useful calculation Can Table 2.4 be simplified further by pooling more rows or columns? Perhaps it might be, but there may well be a risk of losing some important or relevant information. So, before considering any further simplification, we shall look at adding information to the table, in the form of the
1.2.1 Data sets in different tabular forms In much of your statistical work, you will begin with data set, often presented in the form of a table, and use the information in the table to produce diagrams and/or summary statistics that help in the interpretation of the data set. However, in practice, much interpretation of data sets can be done directly from an appropriate table of data, or by re-presenting the data in a rather different tabular form. Dealing with data in tables is the subject of this section and the next. By the time
1.2.8 Problems with graphics: nuclear power stations
Figure 8 shows a pie chart of the data on nuclear power stations from Table
1.2.6 Problems with graphics In this subsection we consider, briefly, some problems that can arise with certain ways of drawing bar charts and pie charts.
Figure 5 shows what is essentially the same bar chart as Author(s):
1.1.4: Infants with SIRDS The data in Table 3 are the recorded birth weights of 50 infants who displayed severe idiopathic respiratory distress syndrome (SIRDS). This is a serious condition which can result in death. 5.2 Doing and undoing Now try the following. Think of a number. Add 4. If my answer is 11, can you work out what number I was thinking of? You might have said ‘What number do I have to add on to 4 to get 11?’ or perhaps ‘If I take away 4 from 11 what number do I get?’ In both cases you should have arrived at the answer 7. In the second method ‘subtracting 4’ undoes the ‘adding 4’ in the original instructions. This process can be illustrated by a ‘doing–undoing diagram’ 3.3 Maths in healthcare The body mass index (BMI) is sometimes used to help determine whether an adult is under- or overweight. It is calculated as follows: Although care needs to be taken in interpreting the results (for example, the formula isn't appropriate for children, old people or those w 3.2 Maths in cookery The time taken to cook a fresh chicken depends on its mass, as given by the following formula: Roughly how long will a chicken with a mass of 2.2 kg take to cook? To use the formula, you need to substitute the mass of the chicken into the right-hand side of the equ 3.1 Introduction In the last section, we considered how a formula could be built up and then how it could be used. This section considers some more complicated formulas, which have already been developed and are used in a variety of different situations – cookery, healthcare, business and archaeology. We hope that these examples illustrate some of the very broad applications of maths and how mathematical relationships can be used in making decisions. As you work through these examples, you may like to consi 1 Exploring patterns and processes Suppose you are tiling a bathroom or kitchen and the last row of square tiles is to be a frieze made up of blank tiles and patterned tiles as shown below. Learning outcomes After completing this unit you should be able to: visualise problems using pictures and diagrams; recognise patterns in a variety of different situations; use a word formula to help solve a problem; derive simple word formulas of your own, for example for use in a spreadsheet; use doing and undoing diagrams to change formulas round; solve problems involving direct and inverse proportion; in Introduction Patterns occur everywhere in art, nature, science and especially mathematics. Being able to recognise, describe and use these patterns is an important skill that helps you to tackle a wide variety of different problems. This unit explores some of these patterns ranging from ancient number patterns to the latest mathematical research. It also looks at some useful practical applications. You will see how to describe some patterns mathematically as formulas and how these can be used to solve pro Acknowledgements Except for third party materials and otherwise stated (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence The audio extracts are taken from M208 Pure Mathematics. © 2006 4.9 Further exercises Determine the equation of the circle with centre (2, 1) and radius 3. The equation of the circle is ( 4.8 General equation of a conic You have already met the parabola, ellipse and hyperbola. So far, you have considered the equation of a conic only when it is in standard form; that is, when the centre of the conic (if it has a centre) is at the origin, and the axes of the conic are parallel to the x- and y-axes. However, most of the conics that arise in calculations are not in standard form. We have seen that any circle can be described by an equation of the form 4.7 Rectangular hyperbola (e = √2) If the eccentricity e of a hyperbola is equal to √2, then e2 = 2 and b = a. Then the asymptotes of the hyperbola have equations y = ±x, so they are at right angles. A hyperbola whose asymptotes are at right angles is called a rectangular hyperbola.
4.6 Hyperbola (e > 1) A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0; the directrix d is the line with equation x = a/e.
4.5 Ellipse (0 < e < 1) An ellipse with eccentricity e (where 0 < e < 1) is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d. We obtain such an ellipse in standard form if the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0; the directrix d is the line with equation x = a 4.4 Parabola (e = 1) A parabola is defined to be the set of points P in the plane whose distances from a fixed point F are equal to their distances from a fixed line d. We obtain a parabola in standard form if the focus F lies on the x-axis, and has coordinates (a, 0), where a > 0; the directrix d is the line with equation x = −a. Thus the origin lies on t 4.3 Focus–directrix definitions of the non-degenerate conics Earlier, we defined the conic sections as the curves of intersection of planes with a double cone. One of these conic sections, the circle, can be defined as the set of points a fixed distance from a fixed point. Here we define the other non-degenerate conics, the parabola, ellipse and hyperbola, as sets of points that satisfy a somewhat similar condition. These three non-degenerate conics (the parabola, ellipse and hyperbola) can be defined as the set of points P in
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Exercise 58
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