Calculate the mass of wood making a closed rectangular box.

Given:
- The wood density is \(0.4 \, g/cm^3\).
- The wood thickness is \(1.5 \, cm\).
- The external dimensions of the box are \(30 \, cm \times 24 \, cm \times 20 \, cm\).

Provide your answer in kilograms.

[tex]\[
\begin{array}{l}
\sqrt{12144+25}=\sqrt{1694}=13 \\
P=4 \times 5=4 \times 13 \\
=52
\end{array}
\][/tex]

3 marks



Answer :

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].