Question

A rectangular prism has a length of 12 in., a width of 5 in., and a height of 414 in.

The prism is filled with cubes that have edge lengths of 14 in.

How many cubes are needed to fill the rectangular prism?

Answer 1

There are needed 9 cubes.

Explanation:

The volume of a rectangular prism is

`V=l * h * w`

Where
`l` is the length,
`h` is the height and
`w` is the width.

In this case, we have

`l=12in\\h=414in\\w=5in`

Replacing this values we can find the volume

`V=12in (5in)(414in)=24,840 in^(3)`

Now, the cubes needed have edge lengths of 14 in, that means their volume is

`V_(cube) = l^(3) =(14in)^(3)=2,744 in^(3)`

To know how many cubes are needed, we just need to divide

`n=(24,840)/(2,744) \approx 9`

Therefore, there are needed 9 cubes.

Answer 2

16320 cubes are needed to fill the rectangular prism.

Explanation:

Given : A rectangular prism has a length of 12 in., a width of 5 in., and a height of
`4(1)/(4)` in.

The prism is filled with cubes that have edge lengths of
`(1)/(4)` in.

To find : How many cubes are needed to fill the rectangular prism?

Solution :

A rectangular prism has

Length of 12 in.

Width of 5 in.

Height of
`4(1)/(4)=(17)/(4)` in.

Volume of rectangular prism is
`V=L* B* H`

`V=12* 5* (17)/(4)`

`V=255 in^3`

A cube has length of
`(1)/(4)` in.

Volume of cube is
`V=L^3`

`V=(1)/(4)^3`

`V=(1)/(64)in^3`

Cubes are needed to fill the rectangular prism is

`=\frac{\text{Volume of rectangular prism}}{\text{Volume of cubes}}`

`=(255)/((1)/(64))`

`=255* 64`

`=16320`

Therefore, 16320 cubes are needed to fill the rectangular prism.