You are going to make a chocolate cakefor some friends.

Find out:•how much of each ingredient you need•The oven temperature and cooking time•The size of the cake•The number of friends you can invite.

1

Ingredients:250g plain choc225g butter5 eggs90g caster sugar15ml cocoa powder10ml vanilla essence

Serves: 15Oven temp: Gas mark 3Cake diameter = 24cmCake height = 15cmCooking time: 45 mins

?

The number of portions you can makeis the 5th triangular number.

1 36

2

=1,3,6,10,15

So 15 portions

You need ‘y’ grams of plain chocolate

y is proportional to the cube of x

y = 432 when x =6

Find y when x = 5 to find out how much chocolate you need !

y x 3 y = kx3 Find k

3

432 = 216k k = 2 y =2x3

y = 2 125 = 250 250g of plain chocolate

The amount of butter in grams is the 15th term in this sequence

1 , 4 , 9 , 16 , ……..

Square numbers ~ find 152

4

15 x 15 = 225

So 225g of butter

Simplify to find the number of eggs

87535

35 = (5 7) = 5 7 = 5 7

5

875 = 5 7 5 5

875 = 5 7 5 5 = 5 5 = 535 5 7

So 5 eggs

Or note that 875 = 35 25 so 875/35 = 25 then result follows.

A triangle

Three sides of a triangle measure:15cm , 36cm and 39cm.

What is the largest angle ?

The size of this angle in degrees is thenumber of grams of caster sugar you need.

Pythagoras’ theorem

6

39

36

15

152 + 362 = 392

225 + 1296 = 1521

(This is a 5, 12, 13 triangle!)

So largest angle = 90So you need 90g caster sugar

Ingredients List:•Plain chocolate ~ 250g•Butter ~ 225g•Eggs ~ 5•Caster sugar ~ 90g•Cocoa powder ~ 15ml•Vanilla essence ~ 10ml

…but how much of each do you need?

7

A

B

C

O

Angle ABC (in°) is the amount of cocoa in mlCircle centre = OAngle OAC = 75

Cocoa !

The angle at the centre is twice the angle at the circumference

8

75 75

30

15

AO = OC = radiiAOC = isos triangleOAC = OCA = 75so AOC= 180-2(75) = 30ABC is half AOC =15 so 15ml cocoa needed

Vanilla essence

The base of this pyramid is a regular hexagon with side lengths of 4cm

The volume of this pyramid is 803 cm3

The amount of vanilla essence in ml is the same as the perpendicular heightof the pyramid in cm.

Pyramid volume = 1/3 base area perpendicular heightRegular hexagon = 6 equilateral triangles

9

Area of one triangle = 43 (by Pythagoras)Area of six triangles = area of base = 243 1/3 area of base = 83 Vol = 803 = 1/3 base area heightSo height = 10cmSo 10ml of vanilla essence is needed. 2 2

44h

h= (42-22)h= (16-4)h= 12h= 4 3h= 23 Area = 2 23

x is the gas mark temperature for baking the cake

(you are looking for the positive solution…obviously)((otherwise the cake wouldn’t cook!))

2 + 1 = 2(x+2) + 1(x+1) = 7 x+1 x+2 (x+2) (x+1) 10

= 3x + 5 = 7 (x+2)(x+1) 10See card 10a!

10

2 + 1 = 7 x + 1 x + 2 10

10a

Algebraic fractions …continued!

Now ‘cross multiply’

10(3x + 5) = 7(x + 1)(x + 2)30x + 50 = 7(x2 + 3x + 2)30x + 50 = 7x2 +21x + 14)0 = 7x2 - 9x – 360 = (7x + 12)(x – 3)

So x = -12/7 or 3

You need gas mark 3 !!

The surface area of the baking tin (sides and base) is 504 cm2

The height of the tin is 15cm

15cm

Find the diameter of the cake

h

2r

+ r2

11

Solution see 11a

11a

Surface area = r2 + 2rh (area of circular base plus area of ‘wrap around’ rectangle)

SA = 504 = r2 + 2r 15504 = r2 + 30r 0 = r2 + 30r – 5040 = (r + 42)(r – 12) r = -42 or 12

So radius = 12cm and diameter = 24cm (size of cake = 24cm diameter, 15cm height)

x is the cooking time in minutes

sin x = 1 2

1

1

45

45

SOH CAH TOA

12

sin 45° = 1 2

2

By Pythagoras

So the cooking time is 45 mins