# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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370 QUOTIENT

Questions must be chosen with care in order not to get a biased result. For example, if you wanted to prevent supermarkets selling alcohol, you might set a question that may produce a negative answer, such as “Do you think supermarkets should be allowed to sell spirits?” Some people might agree to the sale of beer or wine, but object to whisky and gin. Another negative response might be obtained to the question “Do you think supermarkets should sell alcohol at all times ?” Many people might generally agree to supermarkets selling alcohol, but make an exception for Sunday.

References: Bias, Census, Sample, Survey.

QUOTIENT

Reference: Dividend.

R

RADIAN

A radian (rad) is a unit used for measuring the size of an angle. A radian is defined in the following way (see figure a). Suppose we have a circle of center O and radius one unit. A sector AOB is drawn that has an arc length also of one unit, as shown in the circle on the left-hand side of the figure. The angle AOB is defined to be one radian. Instead of the arc length AB being 1 unit, suppose the chord length AB is 1 unit, as shown in the right-hand side of the figure. The angle AOB in the circle on the right-hand side of the figure is 60° because triangle AOB is an equilateral triangle. Comparing the two figures we can see that one radian is slightly smaller than 60°.

A

/ v \ / V VsA

( /1 racQ q { /§o3[

I / I

(a)

There are a little over six radians in a full turn of 360°. The exact number of radians in a full turn is obtained by dividing the circumference (C) of a circle of radius one unit by an arc length of one unit:

C = it x diameter The formula for the circumference of a circle

C — ix x l The diameter of the circle is two units

2jt 1 = 2tt Circumference arc length

There are 2tt radians in a full turn of 360°. We also write

1 radian = 360° 2jt

= 57.29577951... degrees This number cannot be written as an

exact decimal

1 radian = 57.3° (to 1 dp)

371

372 RADIAN

Conversions To convert degrees to radians multiply the number of degrees by jr/180. To convert radians to degrees multiply the number of radians by 180/jr.

Example 1. (a) Convert 3.2 radians to degrees, (b) Convert 94.6° to radians. Solution. For (a), write 180

3.2 x = 183.3 (to 1 dp) Using conversion formula and then the

71 calculator

For (b), write

71

94.6 x -------- = 1.65 rad (to 2 dp) Using the conversion formula and then the

^ calculator

Radians are used to solve problems concerning arc length (s) and area of a sector (A); see figure b, where the appropriate formulas are stated.

Example 2. Amanda is schooling up her horse George for a show. He is trotting in a circle at the end of a rope held by Amanda, who is at point A in figure c. The length of the rope is 10 meters.

(a) As Amanda turns through an angle of 2 radians, find the distance trotted by George along the minor arc of the circle from G\ to G2.

(b) Find the area of the minor sector swept out by the rotation of the rope.

RANDOM NUMBERS 373

Solution, (a) The arc length is the distance Gj to G2 measured around the circumference of the circle. Write

G1G2 = 10 x 2 Substitute R = 10, 0 = 2 into the formula s = R0 GiG2 = 20 meters

(b) For the area, write

A = | x 102 x 2 Substituting R = 10 and 6=2 into the formula A = \R26 A = 100 cm2

If the angle is in degrees, it must be converted into radians in order to use these formulas for arc length and area of a sector.

References: Arc, Arc Length, Area, Circumference.

RADIOACTIVE DECAY

Reference: Exponential Decay.

RADIUS

References: Chord, Circle.

RANDOM NUMBERS

Random numbers are numbers that occur without any design or pattern. They are numbers that occur completely by chance. For example, when you roll a die all of the numbers 1,2, 3,4,5, and 6 have an equal chance of turning up, provided the die is not biased. A die will only generate random numbers from 1 to 6. If you wanted random numbers from 1 to 9 there are various options open to you. One option is to place the numbers 1 to 9 in a hat and draw one number at a time without looking. Of course after each number is drawn it would have to be replaced, because at each draw every one of the numbers must have an equal chance of being drawn from the hat. Another option is to design a “spinner” with the numbers 1 to 9 equally spaced, as in the figure. A

374 RANGE

scientific calculator can generate random numbers of any desired size, and the reader is referred to the handbook for instructions for the particular calculator they use.

Random numbers are used in statistics to select a random sample from a population, as explained in the following example. One day the Principal strides into Luke’s classroom and decides to inspect the students’ books to check them for neatness. He does not want to read through all the books, because he is very busy, so he decides to select just five books at random. There are various options. The first option is to choose the first five students he meets on entering at the back of the room. But he knows that lazy students may sit at the back and these books might be the worst in the class. The second option is to choose the 1st, the 6th, the 12th, the 18th, the 24th, and the 30th names on the roll. This option will give a random sample. The third option, which is also a random selection, is explained below.

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