Let's break down the solution step-by-step:
1. Determine the external measurements of the box:
- Length = 30 cm
- Width = 24 cm
- Height = 20 cm
2. Determine the thickness of the wood:
- Thickness = 1.5 cm
3. Calculate the internal dimensions of the box:
- Internal length = External length - 2 × Thickness
[tex]\[
30 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 30 \, \text{cm} - 3 \, \text{cm} = 27 \, \text{cm}
\][/tex]
- Internal width = External width - 2 × Thickness
[tex]\[
24 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 24 \, \text{cm} - 3 \, \text{cm} = 21 \, \text{cm}
\][/tex]
- Internal height = External height - 2 × Thickness
[tex]\[
20 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 20 \, \text{cm} - 3 \, \text{cm} = 17 \, \text{cm}
\][/tex]
4. Calculate the volumes:
- Volume of the external box:
[tex]\[
\text{Volume}_{\text{external}} = \text{Length}_{\text{external}} \times \text{Width}_{\text{external}} \times \text{Height}_{\text{external}}
\][/tex]
[tex]\[
= 30 \, \text{cm} \times 24 \, \text{cm} \times 20 \, \text{cm} = 14400 \, \text{cm}^3
\][/tex]
- Volume of the internal box:
[tex]\[
\text{Volume}_{\text{internal}} = \text{Length}_{\text{internal}} \times \text{Width}_{\text{internal}} \times \text{Height}_{\text{internal}}
\][/tex]
[tex]\[
= 27 \, \text{cm} \times 21 \, \text{cm} \times 17 \, \text{cm} = 9639 \, \text{cm}^3
\][/tex]
5. Calculate the volume of the wood used to make the box:
- Volume of the wood:
[tex]\[
\text{Volume}_{\text{wood}} = \text{Volume}_{\text{external}} - \text{Volume}_{\text{internal}}
\][/tex]
[tex]\[
= 14400 \, \text{cm}^3 - 9639 \, \text{cm}^3 = 4761 \, \text{cm}^3
\][/tex]
6. Determine the density of the wood:
- Density = 0.4 g/cm³
7. Calculate the mass of the wood:
- Mass of the wood (in grams):
[tex]\[
\text{Mass}_{\text{wood}} = \text{Volume}_{\text{wood}} \times \text{Density}
\][/tex]
[tex]\[
= 4761 \, \text{cm}^3 \times 0.4 \, \text{g/cm}^3 = 1904.4 \, \text{g}
\][/tex]
- Convert the mass to kilograms:
[tex]\[
1904.4 \, \text{g} \div 1000 = 1.9044 \, \text{kg}
\][/tex]
Hence, the mass of the wood making the box is [tex]\(1.9044 \, \text{kg}\)[/tex].
