Download Maths Extended - Chapter 8 - Pearson-Global

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
Formulae is the plural of formula.
You will need to know
l the correct order of operations
Multiplication
Indices
Brackets
Division
"
"
Addition
" Subtraction
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 5 500 2 45x
$5 5 500c
The melons cost 45c each so the
cost, in cents, for x melons is 45x.
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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 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 20 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 2.
Write down a formula for the perimeter, p, of this rectangle.
3x + 1
x+2
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 2
If a 5 5, b 5 4 and c 5 3 work out the value of these expressions.
a 1 3
(b) 3b2 2 1 (c) 5c 1 1
(a)
2
2
The dividing line acts like a bracket.
You must work out the numerator
first.
a 1 3 5 5 1 3
2
2
5 8 4 2
5 4
(a)
8
Remember 2 5 8 4 2.
(b) 3b 2 1 5 3 × 4 2 1
5 3 × 16 2 1
5 48 2 1
5 47
2
2
continued .
You must work out the indices first
(42 5 16), then the multiplication
(3 × 16), then
the subtraction (48 2 1).
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Formulae
(c) 5c 1 1 5 5 × 3 1 1
b
4
5 15 1 1
4
16
5 4
5 4
EXERCISE 8B
1 If r 5 5, s 5 4 and t 5 3, work out the value of these expressions.
r13
s15
t17
(a)
(b)
(c)
2
3
2
(e) 4t2 2 6
(f) 2s2 1 r
(d) 3r2 1 1
(g) 4(5s 1 1)
(h)t(r 1 s)
(i) 5(2s 2 3t)
5t 1 1
4r 2 2
3s 1 t
(j)
(k)
(l)
s
t
r
3r2 5 3 × r2 5 3 × r × r
2 Copy and complete this table.
x
1
2
x2 1 2x
3
4
5
3.7
3.8
32 1 2 × 3 5 9 1 6 5 15
15
3 Copy and complete this table.
x
3
x 2 x
3
4
3.5
Remember x3 5 x × x × x
43 2 4 5 64 2 4 5 60
60
4 If A 5 6, B 5 24, 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) A2 1 2B 1 C
2A 1 3
4B 1 D
A2 1 3B
(d)
(e)
(f)
C
2
C
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Substituting into formulae
EXAMPLE 3
A formula for working out acceleration is
v2u
a 5 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.
v2u
t
50 2 10
5 8
5 40 4 8
5 5
a 5 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 21 at2
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 21 at2
5 3 × 5 1 21 × 8 × 52
5 15 1 4 × 25
5 15 1 100
Do 52 5 25 first;
then 3 × 5 5 15 and 12 × 8 5 4;
then 4 × 25 5 100;
then 15 1 100 5 115.
5 115
EXERCISE 8C
v2u
to work out the value of a when
t
(a) v 5 15, u 5 3, t 5 2
(b) v 5 29, u 5 5, t 5 6
(c) v 5 25, u 5 7, t 5 3
(d) v 5 60, u 5 10, t 5 4.
1 Use the formula a 5 120 Algebra
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Formulae
2 A person’s Body Mass Index, b, is calculated using the formula
m
b 5 2
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 5 1.8
(b) m 5 38, h 5 1.4
(c) m 5 85, h 5 1.9
(d) m 5 59, h 5 1.7.
a
3 The formula for the area of a trapezium is
A 5 12 (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 5 6.
h
b
4 Use the formula s 5 ut 1 12 at2 to work out the value of s when
(a) u 5 3, a 5 10, t 5 2
(b) u 5 7, a 5 6, t 5 5
(c) u 5 2.5, a 5 5, t 5 4
(d) u 5 24, a 5 8, t 5 3.
5 A formula for working out the velocity of a car is
v 5 √u2 1 2as
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 2
(d) u 5 7, a 5 4, s 5 15.
6 A formula for the surface area of a cone, including the base, is
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 2 cm, l 5 13 cm
(b) r 5 4 cm, l 5 10 cm.
Give your answers to 3 s.f.
surface area of cone 5 pr(r 1 l)
l
r
7 A formula for the total surface area of a cylinder is
where r is the radius and h is the height.
Work out the surface area of a cylinder with these dimensions.
(a) r 5 3.7 cm, h 5 10.8 cm (b) r 5 4.2 cm, h 5 4.1 cm.
Give your answers to 3 s.f.
surface area of cylinder 5 2pr(r 1 h)
r
h
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Using formulae
EXAMPLE 5
The perimeter of a rectangle is given by P 5 2l 1 2w, where l is the
length and w is the width.
Work out the value of l when P 5 24 and w 5 5.
P 5 2l 1 2w
24 5 2l 1 2 × 5
24 5 2l 1 10
Substitute the values you know, of
P and w, into the formula, then
solve the equation to find l.
24 2 10 5 2l
14 5 2l
7 5 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 12 and l 5 4
(b) A 5 36 and l 5 9
(c) A 5 42 and l 5 7
(d) A 5 60 and l 5 15.
l
w
2 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 2.
(b) The value of R when V 5 35 and I 5 5.
(c) The value of I when V 5 240 and R 5 30.
(d) The value of I when V 5 240 and R 5 40.
3 The perimeter of a rectangle is given by
l
P 5 2l 1 2w
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 32 and w 5 7.
(c) find w when P 5 60 and l 5 17.
(d) find w when P 5 50 and l 5 13.5.
w
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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 20, a 5 7, t 5 4.
8.2 Changing the subject of a formula
You will need to know that
addition and subtraction are inverse operations
multiplication and division are inverse operations
squaring and finding the square root are inverse operations
l
l
l
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 2l 1 2w.
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.
You need to have a on its own on
one side.
Subtract 8 from both sides as you
would when solving an equation.
d 5 a 1 8
d 2 8 5 a
EXAMPLE 7
Make x the subject of the formula y 5 5x 2 2.
Add 2 to both sides to leave 5x on
its own.
y 5 5x 2 2
y 1 2 5 5x
y 1 2 5 x
5
Then divide both sides by 5 to leave
x on its own.
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EXAMPLE 8
Rearrange P 5 4g 1 2h to make g the subject.
P 5 4g 1 2h
P 2 2h 5 4g
P 2 2h 5 g
4
Subtract 2h 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)
Square both sides first.
V 2 5 w 1 y
V 2 2 y 5 w
Then subtract y from both sides.
EXAMPLE 10
q2
Rearrange p 5 2 s to make q the subject.
r
q2
p 5 2 s
r
q2
p 1 s 5 r
r (p 1 s) 5 q2
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.
EXERCISE 8e
1Rearrange these formulae to make y the subject.
(a) x 5 5y 2 6
(b) x 1 2y 5 8
(c) x 1 3y 2 8 5 0
(d) 2x 1 5y 2 7 5 0
(e) x 2 y 5 4
(f) 2x 2 5y 5 20
2Make x the subject of these formulae.
(a) y 5 5x 2 6
(b) y 5 12 x 2 8
(c) y 1 12 x 5 2
(d) y 2 x 5 7
(e) 2y 1 x 5 9
(f) 2y 1 5x 5 9
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Formulae
3Make r the subject of these formulae.
(a) p 5 3(4r 1 5t)
(b) v 5 5(7r + h)
3r 2 2s
6p 2 5r
(c) w 5
(d) y 5
5
8
4Rearrange these formulae to make a the subject.
(b) b 5 12 a 1 7
(a) b 5 12 a 1 6
(c) b 5 13 a 2 1
(d) b 5 14 a 2 3
(e) b 5 2(a 1 1)
(f) b 5 3(a 2 5)
Multiply 12 a by 2 to get a.
5Rearrange these formulae to make x the subject.
(a) 3(x 1 y) 5 5y (b) 2(x 2 y) 5 y 1 5
x
2x
(c) z 5 2 5
(d) 3p 5 2 s
y
q
6Rearrange these formulae to make w the subject.
(b) A 5 √w 2 a
(a) K 5 √w 1 t
(c) h 5 2√w 1 l
(d) T 5 √wr 1 5
Use Example 9 to help.
7Rearrange these formulae to make r the subject.
r2
r2
(a) t 5 2 m
(b) h 5 1 3a
g
4
(d) A 5 p(r2 2 s2)
(c) V 5 13 pr2h
Use Example 10 to help.
8A formula for the total surface area of a cylinder is
A 5 2pr(r 1 h)
where r is the radius and h is the height.
Rearrange the formula to make h the subject.
9A formula for the period of a pendulum is
l
T 5 2p
g
Rearrange the formula to make l the subject.
The period T is the time for one
complete swing; l is the length and
g is a constant.
10The formula F 5 1.8C 1 32 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.
11A formula is given as v2 5 u2 + 2as.
(a) Find v when u 5 10, a 5 4 and s 5 12.
(b) Rearrange the formula to make a the subject.
(c) Rearrange the formula to make u the subject.
12A formula is given by
h2 1 g2
T 5 
g2
(a) Find T when g = 10 and h = 2.
(b) Rearrange the formula to make h the subject.
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EXAMINATION QUESTIONS
1
Make h the subject of the formula g 5 √h 1 i.
[2]
(CIE Paper 2, Nov 2000)
2
Make y the subject of the formula x 5
4 1 √y
.
3
[3]
3x
1 5.
2
[3]
(CIE Paper 2, Jun 2001)
3
Make x the subject of the formula y 5
(CIE Paper 2, Nov 2001)
4
5
Make V the subject of the formula T = V 1 1 .
[3]
(CIE Paper 2, Jun 2002)
5
The surface area of a person’s body, A square metres, is given by the formula
hm
A 5 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.
(b) Erik has a mass of 80 kg. Find his height if A 5 1.99.
(c) Make h the subject of the formula.

[1]
[2]
[3]
(CIE Paper 4, Nov 2003)
6
Make c the subject of the formula √3c 2 5.
[3]
(CIE Paper 2, Nov 2004)
126 Algebra
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