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

ISBN 0-471-15045-2

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72

CHANGING THE SUBJECT OF A FORMULA

measure time. This system of units was replaced in 1960 by the SystÃ¨me International dâ€™UnitÃ©s, or SI units. The SI units use meters (m), kilograms (kg), and seconds (s) as the basic units of length, mass, and time, respectively. Other units in this system are multiples, and fractions, of 1000 of these basic units. Exceptions to this rule of multiples (and fractions) of 1000 are 10 millimeters = 1 centimeter, 100 centimeters = 1 meter, 10000 square meters = 1 hectare, and 100 hectares = 1 square kilometer.

References: Centimeter, Gram, Hectare, Kilogram, Meter, Metric Units, Millimeter, Second, SI Units.

CHANCE

Chance is another name for probability. Reference: Probability.

CHANGING THE SUBJECT OF A FORMULA

The formula to find the area of a circle if the radius is known is A = ttR2. The subject of this formula is A, the term on the left-hand side of the equals sign. Suppose you knew the value of the area of the circle and wanted to calculate its radius. In other words, you need a formula where R is the subject. The process of rearranging a formula to make another variable the subject is called changing the subject of a formula.

The rules for changing the subject of a formula are the same as those for solving an equation. The first step of changing the subject of a formula is to rewrite the formula with the term that will be the subject of the formula on the left of the equals sign, as illustrated in the following examples.

Example. The formula for the area of a circle is A = ttR2. Find the radius of the circle if the area is 5.57 cm2.

Solution. The first step is to make R the subject of the formula:

A = TtR2

TtR2 = A Rewriting with the term in R on the left of the equals sign

A

R2 = â€” Dividing both sides by it to isolate the term R2

R = Â±

Taking the square root of both sides of the equation

CHANGING THE SUBJECT OF A FORMULA

73

R

R

5.57

71

Discarding the Â± sign, because radius cannot be negative

Substituting A = 5.57

R = 1.33 cm (to 3 sf) Using the calculator value for it

The answer is rounded to three significant figures, since A = 5.57 cm2 was given to 3 sf in the question.

Example. When she was young, Isabelâ€™s grandmother had a formula she quoted to her at bedtime. The number of hours sleep young children need is given by the rale

H =7 +

19

where H is the number of hours of sleep and y is the age of the child. Rearrange the formula to make y the subject.

Solution. Write

7 + 19

7 +

o Rewriting with the term in y on the

z left of equals sign

-;y 7 = H 7 Subtracting 7 from both sides of the

z equation

19 - y y = H 7 2 Simplifying 7â€”7 = 0

r~7 1 g, II 1 ON Tâ€”H Multiplying both sides by 2, and using brackets

ON Tâ€”H 1 r~7 1 & II 1 Subtracting 19 from both sides

ON Tâ€”H 1 r-H 1 <N II 1 Expanding the brackets

-y = 2H- 33 -14-19 = -33

y = â€”2 H + 33 Multiplying both sides by â€” 1, which

changes the signs

Note. It is obvious that grandmotherâ€™s formula only works for children. For example, a person aged 50 would need â€”8.5 hours of sleep!

References: Balancing an Equation, Formula, Perimeter, Rounding, Solving an Equation, Square Root.

74 CIRCLE

CHORD

A chord is a straight line segment joining two points on a curve. The curve we will study is the circle. In figure a, AB is a chord of the circle. The perpendicular bisector, or mediator, of the chord of a circle always passes through the center of the circle.

(a) (b)

The longest possible chord, which passes through the center O of the circle, is called the diameter (see figure b).

When one or both ends of a chord are extended the resulting line is called the secant of the circle. Two examples are shown in figure c.

(c)

References: Center of a Circle, Circle, Mediator, Perpendicular Bisector.

CIRCLE

This is the set of points in a plane that are equidistant from a fixed point O (see figure a). The fixed point is called the center of the circle. The concept of a circle is explained here.

(a)

Amanda is schooling up her horse George for a show. She stands in one place, at point 0, holding the end of a rope. The other end of the rope is tied to George, who is at G. As George trots around her, Amanda turns so that she is always facing him, but

CIRCLE GRAPH 75

keeps the rope tight and the same length. George is always the same distance from Amanda, all the positions of George as Amanda does a full turn form a circle. The different parts of the circle are illustrated here in figure b, but more information is available about them under their own separate headings.

Chord

/Diameter \ /segments

1 â€”â€¢C-PÂ®nter | f Sector^ \ }

\ Radius

(b)

If the radius of a circle is R, then the area of the circle is given by the formula A = TtR2. The circumference of the circle is given by the formula C = 2jtR, or C = ttD, where D is the diameter of the circle. The circumference of a circle is its perimeter.

Example. Lukeâ€™s father is going to build him a fishpond in the garden. Luke insists on having a circular pond. To hold 10 fish, the pond must have a diameter of 3 meters. Calculate the circumference and the area of the pond.

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