Question

What is the maximum number of rectangular blocks measuring $3$ inches by $2$ inches by $1$ inch that can be packed into a cube-shaped box whose interior measures 6 inches on an edge?

• A. $24$
• B. $28$
• C. $30$
• D. $36$

Answer 1

The correct option is D $36$

Given:

Dimensions of the Rectangular block are $(l$, $b$, $h)$ $=$ $(3$, $2$, $1)$.

Length of an edge of the cube-shaped box $=$ $6$.

To find the maximum number of such rectangular blocks that can be packed into the above cube-shaped box;

Let ${}^{\prime }{n}^{\prime }$ be the maximum number of rectangular blocks.

Volume of ${}^{\prime }{n}^{\prime }$ rectangular blocks $=$ $n$ $\times$$l$ $\times$ $b$ $\times$ $h$.

$=$ $n$ $\times$$3$ $\times$ $2$ $\times$ $1$

$=$ $6$ $\times$ $n$

Volume of the above cube of side $(s=6)$ = $s^3$ $=$ $6^3$

Equating the volume of the above, we get

$6$ $\times$ $n$ $=$ $6^3$

$\Rightarrow n$ $=$ $6^2$

$\Rightarrow n$ $=$ $36$

Therefore, maximum number of such rectangular blocks that can be packed into the above cube-shaped box is ${}^{\prime }{36}^{\prime }$.