Question

A rectangular prism is completely packed with 63 cubes of edge length fraction 1/3 inch, without any gap or overlap. Which of these best describes the volume of this rectangular prism?
2 unit cubes and 2 smaller cubes of volume fraction 1 over 27 cubic inch each
2 unit cubes and 9 smaller cubes of volume fraction 1 over 27 cubic inch each
4 unit cubes and 2 smaller cubes of volume fraction 1 over 27 cubic inch each
4 unit cubes and 3 smaller cubes of volume fraction 1 over 27 cubic inch each

Answer 1

The volume of the prism is a function of the cubes in it.

The volume of the prism is: (b) 2 unit cubes and 9 smaller cubes of volume fraction 1 over 27 cubic inch each

From the question, we have:

$n = 63$ --- number of cubes

$l = \frac 13$ --- the edge length of each cube

So, the volume of a cube is:

$v = l^3$

This gives

$v = \frac 13^3$

$v = \frac 1{27}$

The volume of the prism is then calculated as:

$V = n \times v$

This gives

$V = 63 \times \frac 1{27}$

$V = \frac {63}{27}$

Rewrite as:

$V = \frac {54 + 9}{27}$

Split

$V = \frac {54}{27} + \frac{9}{27}$

$V = 2 + \frac{9}{27}$

Rewrite as:

$V = 2 + 9 \times \frac{1}{27}$

The above equation means that, the volume of the prism is:

(b) 2 unit cubes and 9 smaller cubes of volume fraction 1 over 27 cubic inch each

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