box of boxes

[algebra]

The Fancy Box Company is designing their new "Box of Boxes" product. It will be one large box containing a number of small boxes that are 4-inch cubes. They advertise that it's a perfect packaging product for a lot of small surprises!
the production manager, needs to figure out how many small boxes will fit in the large box. When the large box is closed, the top and bottom are each 192 square inches. The front and back are each 128 square inches. The sides are each 96 square inches.
What are the dimensions of the large box? How many small boxes will fit in the large box?

 

[soroban]

Hello, algebra!

The Fancy Box Company is designing their new "Box of Boxes" product.
It will be one large box containing a number of small boxes that are 4-inch cubes.
They advertise that it's a perfect packaging product for a lot of small surprises!

The production manager needs to figure out how many small boxes will fit in the large box.
When the large box is closed, the top and bottom are each 192 square inches.
The front and back are each 128 square inches.
The sides are each 96 square inches.

(a) What are the dimensions of the large box?
(b) How many small boxes will fit in the large box?

Let \(\displaystyle L,W,H\) be the length, width, height of the large box.

We are told: \(\displaystyle \;\begin{Bmatrix}LW &=& 192 & (1) \\ LH &=& 128 & (2) \\ WH &=& 96 & (3) \end{Bmatrix}\)

Divide \(\displaystyle (1) \div (3)\!:\;\dfrac{LW}{WH} \,=\,\dfrac{192}{96} \quad\Rightarrow\quad L \,=\,2H\)

Substitute into (2): \(\displaystyle \;2H\!\cdot\!H \,=\,128 \quad\Rightarrow\quad H^2 \,=\,64 \quad\Rightarrow\quad H \,=\,8\)

\(\displaystyle \quad\)Then: \(\displaystyle \:L\,=\,16,\;W\,=\,12\)

(a) Therefore: \(\displaystyle \:\{L,W,H\} \:=\: \{16,12,8\}\)


The volume of the large box is: \(\displaystyle \:16\cdot12\cdot8 \,=\,1536\text{ in}^3\)
The volume of the small cube is: \(\displaystyle \:4^3 \,=\,64\text{ in}^3\)

(b) The large box will contain: \(\displaystyle \:\dfrac{1536}{64}\,=\,24\text{ cubes.}\)