Unit 11 Section 3 : Using Formulae

In this section we make use of formulae and develop simple formulae ourselves. First we begin with some revision of working with negative numbers.

Example 1

If a = 6, b = –5, c = –7 and d = 3, calculate:

(a)

a + c

a + c = 6 + (-7)
= 6 - 7
= -1
(b)

a - b

a - b = 6 - (-5)
= 6 + 5
= 11
(c)

bc

bc = (-5) × (-7)
= 35
(d)

b^2 + cd

b^2 + cd = (-5)^2 + (-7) × 3
= 25 + (-21)
= 25 - 21
= 4

Example 2

A triangle has sides of length x, x + 4 and x + 8, as shown in the diagram.

(a)

Write down a formula for the perimeter, p, of the triangle.

p = x + (x + 4) + (x + 8)
p = 3x + 12
(b)

Calculate the perimeter when x = 10.

p = 3 × 10 + 12
p = 30 + 12
p = 42
(c)

Calculate x when the perimeter is 45.

45 = 3x + 12
33 = 3xSubtracting 12 from both sides
x = Dividing both sides by 3
x = 11

Example 3

A removal firm charges £80 plus £2 for every mile that their removal van travels.

(a)

Write down a formula for the cost, £C, of a move of n miles.

C = 80 + 2n

(b)

Calculate the cost of moving 262 miles.

C = 80 + 262 × 2
C = 80 + 524
C = £604
(c)

A move costs £500. How far did the removal van travel?

500 = 80 + 2n
420 = 2nSubtracting 80 from both sides
n = Dividing both sides by 2
n = 210 miles, so the van travelled 210 miles.

Exercises

Note: To type indeces on this page use ^ sign. e.g.  n2
Question 1

If p = 6, q = –2, r = 12 and s = –5, calculate:

(a)p + q
(b)p - s
(c)r - q
(d)q^2 + s^2
(e)p - r
(f)r + s^2
(g)pq + r
(h)r - qs
(i)2r + 5q
(j)6q - 2p
(k)4s - 2r
(l)3q + 7s
Question 2

The diagram shows a quadrilateral.

(a)

Write down a formula for the perimeter, p, of the quadrilateral.

p =
p = x + (x + 1) + (x + 2) + (x + 2) = 4x + 5
(b)

Calculate the perimeter if x = 14.

p =
(c)

Determine x if the perimeter is 73.

x =
Question 3

The diagram shows a parallelogram.

(a)

Write down a formula for the perimeter, p, of the parallelogram.

p =
(b)

Calculate the perimeter when x = 7.

p =
(c)

Determine x if the perimeter is 182.

x =
Question 4

The diagram shows a kite.

(a)

Write down a formula for the perimeter, p, of the kite.

p =
(b)

Calculate the perimeter if x = 9.

p =
(c)

Determine x if the perimeter is 118.

x =
Question 5

The semi-circle shown has radius r.

(a)

Calculate the area and perimeter if r = 5 cm.

Area = cm²
Perimeter = cm
Area = πr2 ;   Perimeter = πr + 2r
(b)

Determine r if the perimeter is 40 cm.

x = cm
r =
(c)

Determine r if the area is 18 cm² .

x = cm
r =
Question 6

A taxi firm charges £1.80 plus 50p per mile travelled.

(a)

Write down a formula for the cost, C pence, of travelling m miles.

C =
(b)

Calculate the cost of a 3-mile journey.

£
(c)

The charge for a journey is £6. What is the distance travelled?

miles
Question 7

Ahmed runs a baked potato stall at the market. He makes a profit of 40p on each potato he sells but he has to pay £50 each day for the stall.

(a)

Write down a formula for the amount of money, in pounds, Ahmed makes on one day if he sells x potatoes.

Amount made =
(b)

Describe what happens if x = 200.

He makes a of £.
(c)

Describe what happens if x = 100.

He makes a of £.
(d)

How many potatoes must he sell to make a profit of £100 in one day?

Question 8

Records for the weather suggest that on average, it is 22 °C warmer in Miami than in Washington. The average temperature in Miami in °C is M. The average temperature in Washington in °C is W.

(a)

Write down a formula for M in terms of W.

M =
(b)

Write down a formula for W in terms of M.

W =
(c)

Determine M if W is –7.

M =
(d)

Determine W if M is –3.

W =
Question 9

An engineer charges £20 plus £p per hour to repair central heating boilers. At one house a repair takes 3 hours and costs £71.

(a)

Determine the value of p.

p =
(b)

Write down a formula for the cost, £C, of a repair that takes x hours.

C =
(c)

A repair costs £96.50. How long does it take?

hours
Question 10

Alan claims that the two shapes shown have the same area.

(a)

Determine a formula for the area of each shape.

Area of left hand shape =
Area of right hand shape =
Area of left hand shape = (x + 1) × (x + 10) + 1 × (x + 4)
= (x^2 + 11x + 10) + x + 4
= x^2 + 12x + 14
Area of right hand shape = (x + 7) × (2x + 2)
= 2x^2 + 16x + 14
(b)

Is Alan correct?

Alan is .
Question 11

Ice creams are sold as cones or tubs at the Beach Kiosk.
A cone costs 60 pence.
A tub costs 40 pence.

The income (F) in pence of the Beach Kiosk can be calculated from the equation F = 60x + 40y where x is the number of cones sold and y is the number of tubs sold.

(a)

On June 1st 1994, x = 65 and y = 80.

Work out the income.

income = £
F = 60x + 40y = 60 × 65 + 40 × 80 = 3900 + 3200 = 7100 pence = £71
(b)

On June 2nd 1994, F = 4800 and x = 50.

Work out how many tubs were sold.

tubs
F = 60x + 40y  ⇒  4800 = 60 × 50 + 40y
 ⇒  4800 = 3000 + 40y
 ⇒  1800 = 40y
 ⇒  y = 1800 ÷ 40
 ⇒  y = 45
So 45 tubs were sold.
(c)

During the first week of last summer 950 ice creams were sold.
437 of them were tubs.

What percentage of the ice creams sold were tubs?

(d)

Estimate the total income in pounds for the summer of 1995 using the information in the box.

Last summer 14 723 ice creams were sold.
Roughly the same number of ice creams is likely to be sold in the summer of 1995.
The ratio of cones to tubs sold is likely to be about 1 : 1.
The cost of a cone is to stay at 60 pence.
The cost of a tub is to stay at 40 pence.

(i)

Write down the number you will use instead of 14 723.

(ii)

Write down the value you will use for the cost of an ice cream.

£
(iii)

Write down your estimate of the total income for the summer of 1995.

£
Question 12
A robot accelerates at a constant rate. It can move backwards or forwards.
When the robot moves, three equations connect the following:
u its initial speed in m/s
v its final speed in m/s
a its acceleration in m/s²
s the distance travelled in m
t the time taken in seconds
The equations are:
v = u + at v^2 = u^2 + 2as s = ut + at^2
For a journey made by the robot:
u = 0.25 m/s t = 3.5 seconds a = –0.05 m/s
Use the appropriate equation to find:
(a)

The distance travelled.

s = m
s = ut + at2 = (0.25 × 3.5) + (0.5 × (–0.05) × 3.52)
= 0.875 – 0.30625 = 0.56875 m
(b)

The robot's final speed.

v = m/s
v = u + at = 0.25 + ((–0.05) × 3.5) = 0.25 – 0.175 = 0.075 m/s
Question 13

Magic squares were used to tell fortunes long ago in China.
They contain whole numbers starting from 1.
The numbers in each row add up to the magic number.

Eleri's magic square has 3 rows.
The magic number is 15.
The size is 3 × 3.

Tony made a magic square with more rows.
The magic number is 2925.
We do not know the size of his square.

When the size of a magic square is n × n, the magic number is .
if n = 3,   = = = = = 15.
So the formula works for Eleri's 3 × 3 magic square.

Find the size of Tony's magic square. You may use trial and improvement.

Tony's magic square was × .

n 12345678910
magic number 15153465111175260369505
n 11121314151617181920
magic number 67187011051379169520562465292534394010
Question 14

Alan throws a ball to Katie who is standing 20 m away.
The ball is thrown and caught at a height of 2.0 m above the ground.

The ball follows the curve with equation

y = 6 + c(10 – x)2 where c is a constant.

(a)

Calculate the value of c by substituting x = 0, y = 2 into the equation.

c =
y = 6 + c(10 – x)2 ⇒ 2 = 6 + c(10 – 0)2
⇒ 2 = 6 + 100c
⇒ –4 = 100c
c = –0.04

Alan throws the ball to Katie again, but this time the ball hits the ground before it reaches her.
The ball follows the curve with equation y = -0.1(x^2 - 6x - 16)

(b)

Calculate the height above the ground at which the ball left Alan's hand.

x =
The ball left Alan's hand at a height of metres.

When x = 0,   y = -0.1(x^2 - 6x - 16) = -0.1(0^2 - [6 × 0] - 16)
= (–0.1) × (–16) = 1.6