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Question 1078873: Rick has 45 cubes each with a side length of 1cm. He wants to use all his cubes to make a cuboid that has a square base. His friend suggests 1cm X 1cm and a height of 45cm. Find the dimensions of a different cuboid he could use from all his cubes. It should have a square base.
The answer states base 2x5 With a height of 9cm. Saying you must have 1 square number and 2 primes so 2x5x9=90. Where does the 90 comes from ? It says a base of 2x9 but that's not square. With a heigh of 9cm. Can anyone help? Thanks
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! one reference on cuboids:
https://www.mathsisfun.com/geometry/cuboids-rectangular-prisms.html
you have 45 cubes, each with a side length of 1 cm.
you want a cuboid with a square base.
1 cube as the base and 45 cubes as the height will do.
basically, you want to divide the number of cubes total by the number of cubes as the base.
if you get an even number, then that's good.
if you can't, then that's no good.
1 cube as the base give you 45 cubes as the height so that's ok.
next size base could be 2 by 2, but that would take 4 cubes and 45 / 4 is not even.
next size base could be 3 by 3, but that would take 9 cubes and 45 / 9 = 5.
that looks like it'll work.
the dimensions would be 3 by 3 by 5 for a total of 45 square cm.
i believe the answer given in the book is wrong.
2 by 5 by 9 would result in total square cm of 90 which is not right.
that would use 90 squares, as you say.
not only that, the base would not be square.
keep in mind the book is not always right.
some books are better than others.
the best have very few to no errors.
the worst?
some are really bad.
the problems can stem anywhere from the person or persons who wrote the book to the translation of the original problems to the final form.
the problems can be human or technical or both, sometimes just a typo is enough to throw it off.
a general rule that i would use is to find the possible square bases first and then divide 45 by the number of cubes used in the base.
if the number divides evenly, then you have your cuboid.
the height is determined by stacking up the cubes used in the base on top of each other.
base of 1 by 1 requires 1 cube divides into 45 evenly.
height is 45 because 45 times the base of 1 cube requires 45 cubes.
base of 2 by 2 requires 4 cubes does not divide into 45 evenly.
base of 3 by 3 requires 9 cubes divides into 45 evenly.
height is 5 because 5 times the base of 9 cubes requires 45 cubes.
base of 4 by 4 requires 16 cubes does not divide into 45 evenly.
base of 5 by 5 requires 25 cubes does not divide into 45 evenly.
base of 6 by 6 requires 36 cubes does not divide into 45 evenly.
base of 7 by 7 requires 49 cubes which is greater than 45 so you're out of possible choices for the base.
the only bases that work are 1 by 1 and 3 by 3.
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