# I'm having trouble finding the volume of a parallelepipid

I'm having trouble resolving an issue involving the volume of a rectangular parallelepiped.

I know that to find the volume, just multiply the factors, that is the length, width, and height. However, none of the alternatives match the multiplication product.

I believe this has to do with "thickness", but I don't quite understand what it means and what I should do to solve the problem. So I would like to know if anyone can help me with this problem.

Question

A factory sells chocolates in a wooden box, as shown.

The wooden box is shaped like a straight-rectangle parallelepiped whose external dimensions, in centimeters, are indicated in the figure. It is also known that the thickness of the wood, on all its sides, is $$0.5\mathrm{cm}$$.

What is the volume of wood used, in cubic centimeters, in the construction of a wooden box like the one described to pack chocolates?

a) $$654$$
b) $$666$$
c) $$673$$
d) $$681$$
e) $$693$$

What I tried:

$$\text{volume} = \text{length}\cdot x\cdot\text{width}\cdot x\cdot \text{height}$$ $$\text{volume} = 20\mathrm{cm}\cdot x\cdot 8\mathrm{cm}\cdot x\cdot 20\mathrm{cm}$$ $$\text{volume} = 3200 \mathrm{cm}^2$$

In the alternatives, there isn't any that match the value I got, and I believe that's because of that part of the question that talks about the thickness of the wood on each side, but since I'm not aware of that, I don't know what to do. I tried to divide the value found by $$6$$ since a parallelepipid has $$6$$ faces, but I didn't come up with a coherent answer.

• Hint: from the volume of the "full" parallelipiped subtract the volume of the empty space inside. Commented Aug 26, 2021 at 20:08

Volume of Wood = External volume - Internal Volume

External Volume = $$20 \times 8 \times 20 = 3200$$

Internal Volume = $$19 \times 7 \times 19 =2527$$

Therefore,

Volume of Wood = $$3200 - 2527 = 673$$

The volume of wood used will be (Volume of "full" box)-(Volume of empty space left ) .

As the thickness of wood is $$0.5 cm$$ , a length of $$0.5 cm+0.5 cm=1 cm$$ will be reduced from each dimension to get the dimensions of the "empty" space left . Therefore , dimensions of "empty" space are $$(19cm)$$$$(19cm)$$$$(7cm)$$ . Therefore volume of empty space=$$(19*19*7) cm^3$$ , or $$2527 cm^3$$ .

Therefore ,volume of wood used = $$(3200- 2527)cm^3$$ or $$673 cm^3$$.