Formulae. Chapter Formulae

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1 Chapter 8 Formulae This chapter will show you how to write a formula from a problem substitute numbers into expressions and formulae change the subject of a formula 8.1 Formulae You will need to know l the correct order of operations Brackets " Indices " Multiplication " Addition Division Subtraction Formulae is the plural of formula. A formula is a general rule that shows how quantities (or variables) are related to each other. For example, v 5 u 1 at This is a formula that shows the relationship between an object s final velocity, v, its initial velocity, u, its acceleration, a, and the time it has been moving, t. Deriving formulae When solving a problem, it often helps to write a formula to express the problem. Start by deciding on a letter to represent an unknown value. EXAMPLE 1 Alex buys x melons. Each melon costs 45 cents. Alex pays with a $5 note. Write a formula for the change, C, in cents, Alex should receive. x 5 number of melons C x $ c The melons cost 45c each so the cost, in cents, for x melons is 45x. Algebra 117

EXERCISE 8A 1 Nilesh buys y mangoes. Each mango costs 48 cents. Nilesh pays with a $5 note. Write a formula for the change, C, in cents, Nilesh should receive.

2 EXERCISE 8A 1 Nilesh buys y mangoes. Each mango costs 48 cents. Nilesh pays with a $5 note. Write a formula for the change, C, in cents, Nilesh should receive. Apples cost r cents each and bananas cost s cents each. Sam buys 7 apples and 5 bananas. Write a formula for the total cost, t, in cents, of these fruit. 3 To cook a chicken you allow 45 minutes per kg and then a further 0 minutes. Write a formula for the time, t, in minutes, to cook a chicken that weighs w kg. 4 To cook lamb you allow 30 minutes plus a further 65 minutes per kg. Write a formula for the time, t, in minutes, to cook a joint of lamb that weighs w kg. 5 A rectangle has a length of 3x 1 1 and a width of x 1. Write down a formula for the perimeter, p, of this rectangle. x + 3x + 1 Substitution This section shows you how to use substitution to find the values of different algebraic expressions. Use mathematical operations in the correct order when substituting values into an algebraic expression. EXAMPLE If a 5 5, b 5 4 and c 5 3 work out the value of these expressions. a 1 3 (a) (b) 3b 1 (c) 5c 1 1 (a) a (b) 3b continued. The dividing line acts like a bracket. You must work out the numerator first. Remember You must work out the indices first (4 5 16), then the multiplication (3 16), then the subtraction (48 1). 118 Algebra

3 Formulae (c) 5c 1 1 b EXERCISE 8B 1 If r 5 5, s 5 4 and t 5 3, work out the value of these expressions. (a) r 1 3 (b) s 1 5 (c) t (d) 3r 1 1 (e) 4t 6 (f) s 1 r (g) 4(5s 1 1) (h) t(r 1 s) (i) 5(s 3t) (j) 5t 1 1 (k) 4r (l) 3s 1 t s t r Copy and complete this table. x x 1 x 15 3r 5 3 r 5 3 r r Copy and complete this table. x x 3 x 60 4 If A 5 6, B 5 4, C 5 3 and D 5 30, work out the value of these expressions. (a) D(B 1 7) (b) A(B 1 1) (c) A 1 B 1 C (d) A 1 3 C (e) 4B 1 D (f) A 1 3B C Remember x 3 5 x x x Algebra 119

4 Substituting into formulae EXAMPLE 3 A formula for working out acceleration is a 5 v u t where v is the final velocity, u is the initial velocity and t is the time taken. Work out the value of a when v 5 50, u 5 10, t 5 8. a 5 v u t Remember the line for division acts like a bracket. You must work out the numerator first. EXAMPLE 4 A formula for working out distance travelled is s 5 ut 1 1 at where u is the initial velocity, a is the acceleration and t is the time taken. Work out the value of s when u 5 3, a 5 8, t 5 5. s 5 ut 1 1 at Do first; then and ; then ; then EXERCISE 8C 1 Use the formula a 5 v u to work out the value of a when t (a) v 5 15, u 5 3, t 5 (b) v 5 9, u 5 5, t 5 6 (c) v 5 5, u 5 7, t 5 3 (d) v 5 60, u 5 10, t Algebra

5 Formulae A person s Body Mass Index, b, is calculated using the formula b 5 m h where m is their mass in kilograms and h is their height in metres. Work out the value of b when (a) m 5 70, h (b) m 5 38, h (c) m 5 85, h (d) m 5 59, h The formula for the area of a trapezium is A 5 1 (a 1 b)h Work out the value of A when (a) a 5 10, b 5 6, h 5 4 (b) a 5 13, b 5 9, h 5 8 (c) a 5 9, b 5 6, h 5 4 (d) a 5 15, b 5 10, h Use the formula s 5 ut 1 1 at to work out the value of s when (a) u 5 3, a 5 10, t 5 (b) u 5 7, a 5 6, t 5 5 (c) u 5.5, a 5 5, t 5 4 (d) u 5 4, a 5 8, t A formula for working out the velocity of a car is v 5 u 1 as where u is the initial velocity, a is the acceleration and s is the distance travelled. Work out the value of v when (a) u 5 3, a 5 4, s 5 5 (b) u 5 6, a 5 8, s 5 4 (c) u 5 9, a 5 10, s 5 (d) u 5 7, a 5 4, s h a b 6 A formula for the surface area of a cone, including the base, is surface area of cone 5 pr(r 1 l) where r is the radius and l is the slant height. Work out the surface area of a cone with these dimensions. (a) r 5 cm, l 5 13 cm (b) r 5 4 cm, l 5 10 cm. Give your answers to 3 s.f. l r 7 A formula for the total surface area of a cylinder is surface area of cylinder 5 pr(r 1 h) where r is the radius and h is the height. Work out the surface area of a cylinder with these dimensions. (a) r cm, h cm (b) r 5 4. cm, h cm. Give your answers to 3 s.f. r h Algebra 11

6 Using formulae EXAMPLE 5 The perimeter of a rectangle is given by P 5 l 1 w, where l is the length and w is the width. Work out the value of l when P 5 4 and w 5 5. P 5 l 1 w 4 5 l l l 14 5 l 7 5 l Substitute the values you know, of P and w, into the formula, then solve the equation to find l. EXERCISE 8D 1 The formula for the area of a rectangle is A 5 lw, where l is the length and w is the width. Work out the value of w when (a) A 5 1 and l 5 4 (b) A 5 36 and l 5 9 (c) A 5 4 and l 5 7 (d) A 5 60 and l The formula for the voltage, V, in an electrical circuit is V 5 IR, where I is the current and R is the resistance. Work these out. (a) The value of R when V 5 18 and I 5. (b) The value of R when V 5 35 and I 5 5. (c) The value of I when V 5 40 and R (d) The value of I when V 5 40 and R The perimeter of a rectangle is given by P 5 l 1 w where l is the length and w is the width. Use the formula to (a) find l when P 5 18 and w 5 4. (b) find l when P 5 3 and w 5 7. (c) find w when P 5 60 and l (d) find w when P 5 50 and l w w l l 1 Algebra

7 Formulae 4 Use the formula v 5 u 1 at (a) to find u when v 5 30, a 5 8, t 5 3 (b) to find a when v 5 54, u 5 19, t 5 7 (c) to find t when v 5 60, u 5 15, a 5 5 (d) to find u when v 5 0, a 5 7, t Changing the subject of a formula You will need to know that l addition and subtraction are inverse operations l multiplication and division are inverse operations l squaring and finding the square root are inverse operations The subject of a formula appears only once, on its own, on one side of the formula. In the formula v 5 u 1 at the variable v is called the subject of the formula. P is the subject of the formula P 5 l 1 w. P is on its own on one side of the formula. You can rearrange a formula to make a different variable the subject. EXAMPLE 6 Rearrange d 5 a 1 8 to make a the subject. d 5 a 1 8 d 8 5 a You need to have a on its own on one side. Subtract 8 from both sides as you would when solving an equation. EXAMPLE 7 Make x the subject of the formula y 5 5x. y 5 5x y 1 5 5x y 1 5 x 5 Add to both sides to leave 5x on its own. Then divide both sides by 5 to leave x on its own. Algebra 13

8 EXAMPLE 8 Rearrange P 5 4g 1 h to make g the subject. P 5 4g 1 h P h 5 4g P h 5 g 4 Subtract h from both sides. Divide both sides by 4. EXAMPLE 9 Rearrange V 5 (w 1 y) to make w the subject. V 5 (w 1 y) V 5 w 1 y V y 5 w Square both sides first. Then subtract y from both sides. EXAMPLE 10 Rearrange p 5 q r s to make q the subject. p 5 q r s p 1 s 5 q r r(p 1 s) 5 q 6 r(p 1 s) 5 q Add s to both sides to leave the term with q on its own. Multiply both sides by r. Square root both sides. 1 Rearrange these formulae to make y the subject. (a) x 5 5y 6 (b) x 1 y 5 8 (c) x 1 3y (d) x 1 5y (e) x y 5 4 (f) x 5y 5 0 Make x the subject of these formulae. (a) y 5 5x 6 (b) y 5 1 x 8 (c) y 1 1 x 5 (d) y x 5 7 (e) y 1 x 5 9 (f) y 1 5x Algebra EXERCISE 8e

9 Formulae 3 Make r the subject of these formulae. (a) p 5 3(4r 1 5t) (b) v 5 5(7r + h) 3r s 6p 5r (c) w 5 (d) y Rearrange these formulae to make a the subject. (a) b 5 1 a 1 6 (b) b 5 1 a 1 7 (c) b a 1 (d) b a 3 (e) b 5 (a 1 1) (f) b 5 3(a 5) 5 Rearrange these formulae to make x the subject. (a) 3(x 1 y) 5 5y (b) (x y) 5 y 1 5 (c) z 5 x y 5 (d) 3p 5 x q s 6 Rearrange these formulae to make w the subject. (a) K 5 w 1 t (b) A 5 w a (c) h 5 w 1 l (d) T 5 wr Rearrange these formulae to make r the subject. (a) t 5 r g m r (b) h a (c) V pr h (d) A 5 p(r s ) 8 A formula for the total surface area of a cylinder is A 5 pr(r 1 h) where r is the radius and h is the height. Rearrange the formula to make h the subject. 9 A formula for the period of a pendulum is T 5 p l g Rearrange the formula to make l the subject. 10 The formula F 5 1.8C 1 3 can be used to convert degrees Celsius, C, to degrees Fahrenheit, F. (a) Convert 15 C to degrees Fahrenheit. (b) Rearrange the formula to make C the subject. 11 A formula is given as v 5 u + as. (a) Find v when u 5 10, a 5 4 and s 5 1. (b) Rearrange the formula to make a the subject. (c) Rearrange the formula to make u the subject. 1 A formula is given by T 5 h 1 g g (a) Find T when g = 10 and h =. (b) Rearrange the formula to make h the subject. Multiply 1 a by to get a. Use Example 9 to help. Use Example 10 to help. The period T is the time for one complete swing; l is the length and g is a constant. Algebra 15

EXAMINATION QUESTIONS 1 Make h the subject of the formula g 5 h 1 i. [] (CIE Paper, Nov 000) Make y the subject of the formula x 5 4 1 y.

[3] 5 The surface area of a person s body, A square metres, is given by the formula A 5 hm 3600 where h is the height in centimetres and m is the mass in kilograms.

10 EXAMINATION QUESTIONS 1 Make h the subject of the formula g 5 h 1 i. [] (CIE Paper, Nov 000) Make y the subject of the formula x y. [3] 3 (CIE Paper, Jun 001) 3 Make x the subject of the formula y 5 3x 1 5. [3] (CIE Paper, Nov 001) 4 Make V the subject of the formula T = (CIE Paper, Jun 00) 5 V 1 1. [3] 5 The surface area of a person s body, A square metres, is given by the formula A 5 hm 3600 where h is the height in centimetres and m is the mass in kilograms. (a) Dolores is 167 cm high and has a mass of 70 kg. Calculate the surface area of her body. [1] (b) Erik has a mass of 80 kg. Find his height if A [] (c) Make h the subject of the formula. [3] (CIE Paper 4, Nov 003) 6 Make c the subject of the formula 3c 5. [3] (CIE Paper, Nov 004) 16 Algebra

Formulae Using an algebraic formula CHAPTER. A h(a b) F 22

Formulae Using an algebraic formula CHAPTER. A h(a b) F 22 Formulae 18 CHAPTER A formula is a way of describing a fact or a rule. A formula can be written using algebraic expressions. A formula must have an sign. In Section 9.6 the area (A) of a trapezium was