Forty-two cubes with $1 \mathrm{~cm}$ edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is $18 \mathrm{~cm}$, then the height, in $\mathrm{cm}$, is
(A) 1
(B) 2
(C) $\frac{7}{3}$
(D) 3
(E) 4
Forty-two cubes with $1 \mathrm{~cm}$ edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is $18 \mathrm{~cm}$, then the height, in $\mathrm{cm}$, is
(A) 1
(B) 2
(C) $\frac{7}{3}$
(D) 3
(E) 4

## Solution 1

Since we have a solid rectangular block with a volume of 42 , its dimensions could be, $42 \times 1 \times 1$ or $6 \times 7 \times 1$ or $21 \times 2 \times 1$ or $2 \times 3 \times 7$ or $14 \times 3 \times 1$.

The only selection which has two factors adding to 9 is $2 \times 3 \times 7$, thus giving the base a perimeter of $2(2+7)=18$ which is required.

So the base is $2 \times 7$ and the height is 3 .

## Solution 2

Since the perimeter of the base is $18 \mathrm{~cm}$, the length and width can only be one of the following:

| $L$ | $W$ |
| :---: | :---: |
| 8 | 1 |
| 7 | 2 |
| 6 | 3 |
| 5 | 4 |

If the height is $h$, these cases lead to the following:

$8 \times 1 \times h=42,7 \times 2 \times h=42,6 \times 3 \times h=42$ or $5 \times 4 \times h=42$.

The only possible value of $h$ which is an integer is $h=3$, with $L=7$ and $W=2$.

ANSWER:(D)