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A cake shaped like a rectangular prism with sides of 9cm by 14cm by 15cm is completely dipped in chocolate
and then cut into small 1 cm^3 cubes. How many of the cubes have chocolate on just one side?
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According to the condition, the entire/(the whole) cake is cut into small (1 cm X 1 cm x 1 cm) cubes.
The total number of these small cubes is 9*14*15 = 1890.
Of them, have chocolate on at least one side those and only those cubes that are in one slice adjacent to some (to any) face of the prism.
The rest of the small cubes HAVE NO chocolate on any small faces.
Those small cubes that have no chocolate on any small face are ALL INTERNAL cubes.
They form smaller prism whose dimensions are 2 cm less than dimensions of the whole prism.
Namely, these dimensions are (9-2) = 7 cm, (14-2) = 12 cm and (15-2) = 13 cm.
So, the number of small INTERNAL cubes is 7*12*13 = 1092.
Thus the number of cubes that have chocolate at least on one side is 1890 - 1092 = 798.
The number of small cubes that have chocolate exactly on one side is
2*(7*12 + 7*13 + 12*13) = 662.
Answer. The number of small cubes that have chocolate at least on one side is 798.
The number of small cubes that have chocolate exactly on one side is 662.
SOLVED.
