Wooden 3

A wooden board 2.5 m long has a cross-section in the shape of a right-angled trapezoid whose parallel sides are 1.2 dm and 8 cm long. The height of the trapezoid is 3 cm. Calculate:

a) the surface area of the board, to determine the amount of stain needed,
b) the weight of the board if the density of the wood is 600 kg/m³,
c) how many such boards can be loaded onto a 3-tonne truck.

Final Answer:

S =  6862.7756 cm²
m =  4.5 kg
n =  666

Step-by-step explanation:

$$\begin{gathered} l=2.5\text{m}\to \text{cm}=2.5\cdot 100\text{cm}=250\text{cm} \\ a=1.2\text{dm}\to \text{cm}=1.2\cdot 10\text{cm}=12\text{cm} \\ c=8\text{cm} \\ v=3\text{cm} \\ {S}_1=\frac{a+c}{2}\cdot v=\frac{12+8}{2}\cdot 3=30{\text{cm}}^2 \\ b=\sqrt{v^2+((a-c)/2{)}^2}=\sqrt{3^2+((12-8)/2{)}^2}=\sqrt{13}\text{cm}\doteq 3.6056\text{cm} \\ o=a+b+c+b=12+3.6056+8+3.6056\doteq 27.2111\text{cm} \\ S=2\cdot S_1+o\cdot l=2\cdot 30+27.2111\cdot 250=6862.7756{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} \rho =600{\text{kg/m}}^3\to {\text{g/cm}}^3=600:1000{\text{g/cm}}^3=0.6{\text{g/cm}}^3 \\ V=S_1\cdot l=30\cdot 250=7500{\text{cm}}^3 \\ {m}_0=\rho \cdot V=0.6\cdot 7500=4500\text{g} \\ m=m_0\to \text{kg}=m_0:1000\text{kg}=4500:1000\text{kg}=4.5\text{kg}=4.5\text{kg} \end{gathered}$$

$$\begin{gathered} m_1=3\text{t}\to \text{kg}=3\cdot 1000\text{kg}=3000\text{kg} \\ n=\lfloor m_1/m\rfloor =\lfloor 3000/4.5\rfloor =666 \end{gathered}$$

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