A raft

I want to build a raft, and I have beams with a square section with side a=20cm and length l=2m, wood density 670 kg/m³. I will connect 10 beams - what is the volume of the raft and its weight? How deep will a raft sink in water (water density 1000kg/m³)? The weight of one person is 50kg. How many people can it carry?

Correct answer:

V =  0.8 m³
m =  536 kg
h =  13.4 cm
x =  5

Step-by-step explanation:

$$\begin{gathered} a=20\text{cm}\to \text{m}=20:100\text{m}=0.2\text{m} \\ l=2\text{m} \\ \rho =670{\text{kg/m}}^3 \\ n=10 \\ w=1000{\text{kg/m}}^3 \\ {m}_1=50\text{kg} \\ b=n\cdot a=10\cdot 0.2=2\text{m} \\ {S}_1=b\cdot l=2\cdot 2=4{\text{m}}^2 \\ V=a\cdot S_1=0.2\cdot 4=0.8{\text{m}}^3 \end{gathered}$$

$m=\rho \cdot V=670\cdot 0.8=536\text{kg}$

$$\begin{gathered} m=m_2 \\ m=w\cdot S_1\cdot h \\ {h}_1=\frac{m}{w\cdot S_1}=\frac{536}{1000\cdot 4}=\frac{67}{500}=0.134\text{m} \\ h=h_1\to \text{cm}=h_1\cdot 100\text{cm}=0.134\cdot 100\text{cm}=13.4\text{cm}=13.4\text{cm} \end{gathered}$$

$$\begin{gathered} m_3=w\cdot V=1000\cdot 0.8=800\text{kg} \\ m+x\cdot m_1=m_3 \\ {x}_1=\frac{m_3-m}{m_1}=\frac{800-536}{50}=\frac{132}{25}=5\frac{7}{25}=5.28 \\ x=\lfloor x_1\rfloor =\lfloor 5.28\rfloor =5 \end{gathered}$$

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