# Additional Mathematics Project

Progressions ± To determine total weight/volume of multi-storey cakes with proportional dimensions, to estimate total ingredients needed for cake-baking, to estimate total amount of cream for decoration.

Part II Best Bakery shop received an order from your school to bake a 5 kg of round cake as shownin Diagram 1 for the Teachers¶ Daycelebration. (Diagram 11)

1)

If a kilogram of cake has a volume of 3800 , and the height of the cake is to be 7.0cm, calculate the diameter of the baking tray to be used to fit the 5 kg cake ordered by your school. = 3.142]

[Use

Answer: Volume of 5kg cake = Base area of cake x Height of cake 3800 x 5 = (3.142)( )² x 7 (3.142) = ( )² 863.872 = ( )² = 29.392 d = 58.784 cm 2)The cake will be baked in an oven with inner dimensions of 80.0 cm in length, 60.0 cmin width and 45.0 cm in height. a)If the volume of cake remains the same, explore by using different values of heights,hcm, and the corresponding values of diameters of the baking tray to beused,d cm. Tabulate your answers

Answer: First, form the formulafor d in terms of h by using the above formula for volume of cake, V = 19000, that is: 19000 = (3.142)(d/2)²h

=

= d²

d=

Height,h (cm) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Diameter,d(cm) 155.53 109.98 89.80 77.77 68.56 63.49 58.78 54.99 51.84 49.18

(b)Based on the values in your table, (i)state the range of heights that isNOTsuitable for the cakes andexplain your answers. Answer: h< 7cm is NOT suitable, because the resulting diameter produced is too large to fit into the oven. Furthermore, the cake would be too short and too wide, making it less attractive.

(ii)suggest the dimensions that you think most suitable for the cake. Givereasons for your answer. Answer: h = 8cm, d = 54.99cm, because it can fit into the oven, and the size is suitable for easy handling.

(c) (i) Form an equation to represent the linear relation betweenhand d. Hence, plot a suitable graph based on the equation that you haveformed. [You may draw your graph with the aid of computersoftware.] Answer: 19000 = (3.142)( )²h

19000/(3.142)h =

= d²

d= d= log d = log d = Log h Log d

log h + log 155.53 0 2.19 1 1.69 2 1.19 3 0.69 4 0.19

(ii) (a) If Best Bakery received an order to bake a cake where the height of the cake is 10.5 cm, use your graph to determine the diameter of the round cake pan required. Answer: h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm (b) If Best Bakery used a 42 cm diameter round cake tray, use yourgraph to estimate the height of the cake obtained. Answer: d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm 3)Best Bakery has been requested to decorate the cake with fresh cream. The thicknessof the cream is normally set to a uniform layer of about1cm (a)Estimate the amount of fresh cream required to decorate the cake using thedimensions that you have suggested in 2(b)(ii). Answer: h = 8cm, d = 54.99cm Amount of fresh cream = VOLUME of fresh cream needed (area x height) Amount of fresh cream = Vol. of cream at the top surface + Vol. of cream at the side surface Vol. of cream at the top surface = Area of top surface x Height of...

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