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Question 1121668: Rosa has a collection of 2000 construction blocks. They measure 2 cm by 5 cm by 4 cm each. Her toy chest has a volume of 50 cm by 24 cm by 75 cm. Is it possible for Rosa to put all her blocks in the toy chest?
Found 4 solutions by solver91311, Boreal, Theo, ikleyn:
Answer by solver91311(24713)   (Show Source):
Answer by Boreal(15235)  (Show Source):
You can put this solution on YOUR website! Match the dimensions
2 cm---with 50 cm 25 blocks
4 cm--with 24 cm 6 blocks
5 cm--with 75 cm 15 blocks
Multiply these, just the way one would compute the volume.
375*6=2250.
Not enough room for all of them.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! she has 2000 construction blocks.
they measure 2 cm by 5 cm by 4 cm each.
her toy chest has a volume of 50 cm by 24 cm by 75 cm.
if you just go by cubic centimeters, then 2000 construction blocks will have a total volume of 2000 * 2 * 5 * 4 = 80,000 cubic centimeters.
the toy chest has a volume of 50 by 24 by 75 equals 90,000 cubic centimeters.
if you just go by volume alone, then it looks like the construction blocks will fit into the toy chest.
however, the shape of the construction blocks matched up against the shape of the toy chest might make it not possible.
one possible arrangement would be to put the longest side of the clocks along the longest side of the toy chest.
the longest side of the toy chest is 75 centimeters.
the longest side of each block is 5 centimeters.
75 / 5 = 15 block that can be laid alongside the longest side of the toy chest.
the next longest side of the toy chest is 50 centimeters.
the next longest side of each block is 4 centimeters.
50 / 4 = 12 blocks that can be laid alongside the next longest side of the boy chest.
those 2 dimensions can hold 15 * 12 = 180 blocks.
the smallest side of the toy chest is 24 centimeters.
the smallest side of each side is 2 centimeters.
24 / 2 = 12 blocks that can be laid alongside the smallest side of the toy chest.
180 * 12 = 2160 blocks.
it looks like the blocks will fit.
there is some wasted space, but not enough to prevent all the blocks from being placed in the toy chest.
they do need to be put in a compatible configuration however.
there are ways in which they are put in the toy chest where you won't be able to get 2000 of them in.
i looked at the possible ways and came up with 4 ways they would fit and 2 ways they wouldn't fit.
how the blocks are place in the toy chest is a definite factors as to whether they will all fit or not.
if they are not lined up properly, there will be wasted space that can't be filled, resulting in not enough space to fit all the blocks.
so, the short answer is yes, they will fit.
the long answer is that it is dependent on how they are placed in the toy chest.
here's the results of the excel spreadsheet that contained my investigation.
the configuration on row 6 and row 8 leaves enough wasted space so that not all the blocks will be able to be fit into the toy chest.
all the others may have some wasted space but not enough to stop all the blocks from being inserted.
Answer by ikleyn(52784)  (Show Source):
You can put this solution on YOUR website! .
I came after our respectful tutors to make my comments.
1. Checking the volumes only (as the tutor @solver91311 suggests) IS NOT ENOUGH for such problems.
It would be enough had the total volume of all 2000 blocks be greater than the toy chest volume, but it is not the case :
the total volume of blocks is 2000*2*5*4 = 80000 cm^3;
the volume of the toy chest = 50*24*75 = 90000 cm^3.
In addition, the way to put the blocks into the toy chest should be checked.
2. Here I'd like to make my correction to the solution by tutor @Boreal.
Match the dimensions
2 cm---with 50 cm 25 blocks
4 cm---with 24 cm 6 blocks
5 cm---with 75 cm 15 blocks
Multiply these, just the way one would compute the volume.
25*6*15*6=2250.
There is ENOUGH room for all of them.
Conclusion : There is ENOUGH room for all 2000 blocks.
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