Day 12: 'Twas the Night Before Christmas
Greetings everyone, read my holiday message here .
Jamie (from the Day 1 problem) is baking some Christmas cakes and cookies for a big Christmas Eve party she is hosting later this evening.
Jamie has made batter for three different types of Christmas treats: gingerbread cake, candy cane crumble, and sticky toffee pudding. Jamie has two baking sheets, one of which is for the gingerbread cake and the candy cane crumble. This baking sheet can hold different baked goods. The other baking sheet is for the leftover gingerbread cake and the sticky toffee pudding. It can hold different baked goods.
If Jamie wants to use up all the space on her baking sheets, and she assumes all of the baked goods to be cubes with a height that is equivalent to the total number of that type of baked good, across all sheets, what is the maximum mass of baked goods that Jamie can make if she knows that the densities of the baked goods are mass unit per unit volume for the gingerbread, mass unit per unit volume for the candy cane crumble, and mass units per unit volume for the sticky toffee pudding?
Give your answer as the absolute value of the difference between the number of gingerbread cakes on the first baking sheet and the number of gingerbread cakes on the second sheet, times 2.
This problem is part of The 12 Days of Math-Mas 2018
The answer is 14.
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1 solution
Let the first sheet contains u gingerbread cakes and y candy cane crumbles, while the second sheet contains x-u gingerbread cakes and z sticky toffee pudding. Then, total mass of the goods is x^3+y^3+8z^3 that we have to maximize, subject to the condition : x+y+z=22+20=42 . This yields x=y=18, and |x-2u|=|3x-40|=14