The barrel

The barrel is 135 cm high. If we fill it to a height of 60 cm, it weighs 163 kg. If we fill it to a height of 80 cm, it weighs 213 kg. How many kilograms weigh a full and how many empty barrels?

Correct answer:

f =  350.5 kg
b =  13 kg

Step-by-step explanation:

$$\begin{gathered} h=135\text{cm} \\ {m}_1=163\text{kg} \\ {m}_2=213\text{kg} \\ {q}_1=\frac{60}{h}=\frac{60}{135}=\frac{4}{9}\doteq 0.4444 \\ {q}_2=\frac{80}{h}=\frac{80}{135}=\frac{16}{27}\doteq 0.5926 \\ {q}_1\cdot a+b=m_1 \\ {q}_2\cdot a+b=m_2 \\ 0.44444444444444\cdot a+b=163 \\ 0.59259259259259\cdot a+b=213 \\ 0.444444a+b=163 \\ 0.592593a+b=213 \\ Pivot:Row1\leftrightarrow Row2 \\ 0.59a+b=213 \\ 0.44a+b=163 \\ Row2-\frac{0.44444444}{0.59259259}\cdot Row1\to Row2 \\ 0.59a+b=213 \\ 0.25b=3.25 \\ b=\frac{3.25}{0.25}=13 \\ a=\frac{213-b}{0.59259259}=\frac{213-13}{0.59259259}=337.5 \\ a=\frac{675}{2}\doteq 337.5 \\ b=13 \\ f=a+b=337.5+13=350.5\text{kg} \end{gathered}$$

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