Question

Small cubes with edge lengths of 1/4 inch will be packed into a rectangular prism (shown below). How many small cubes are needed to completely fill the rectangular prism?

Thank you for the answer!

Answer 1

it's a volume question

Explanation:

First, find the volume of the small shape. `( 1/4 times 1/4 times 1/4) = 1/64`

Then, divide the volume of the bigger shape. `(4.1/2 times 5 times 3.3/4) = 675/8`

Finally, divide the big shape volume by the small shape volume. `(675/8 divide 1/64)= 5400`

Smaller shapes will fit into bigger shapes.

Answer 2

5400

Explanation:

Using a word incorrectly for mathematics; it is a matter of how many 'towers' of `1/4`inch blocks you can fit it.

Consider the height of one 'tower'

Height `->3 3/4" inches "-> 15/4`

`color(brown)("So for the height we have 15 blocks")`

Consider the 5 inches depth `->20/4`

`color(brown)("So for the depth we have 20 'towers'")`

Consider the `4 1/2` width `->18/4`

`color(brown)("So for the width we have 18 'towers'")`

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One approach for working it out without a calculator.
Using a calculator is much easier.

Note that 20 is the same as `10xx2` Multiplying by 10 is easy and doubling is straight forward. Taking advantage of this we have:

Total count of 'towers' `->18xx20 = 18xx10xx2 = 360`

`color(white)()`
`color(white)()`

Total count of blocks at 15 per 'tower' `->15xx360 = 5400`

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What follows is just a way of thinking!

`color(white)("ddddd") 15color(white)("d")->color(white)("d")[10color(white)("dv")color(white)("dd")+color(white)("dddd")5color(white)("ddd"2/2)]`

`color(white)("ddddd")15color(white)("d")->color(white)("d")[10color(white)("dd")color(white)("dd")+ color(white)("d") (1/2xx10)]`

`360xx15->360[10color(white)(/"d")color(white)("d.")+color(white)("d")(1/2xx10)]`

`360xx15 ->[10xx360]+[1/2xx10xx360]`

`color(white)("d")`

`(10xx360)color(white)("dd.")=3600`
`1/2xx10xx360=ul(1800 larr" Add"`
`color(white)("dddddddddddddd")5400`