Floating barrel

Barrel (cylinder shape) floats on water, top of barrel is 8 dm above water and the width of surfaced barrel part is 23 dm. Barrel length is 24 dm.

Calculate the volume of the barrel.

Correct result:

V =  11343.3 dm³

Solution:

$$\begin{gathered} d=24 \\ v=8 \\ a=23 \\ {r}^2=(a\mathrm{/}2{)}^2+(r-v{)}^2 \\ {r}^2=(a\mathrm{/}2{)}^2+r^2-2rv+v^2 \\ 0=(a\mathrm{/}2{)}^2-2rv+v^2 \\ 2rv=(a\mathrm{/}2{)}^2+v^2 \\ r=(v^2+(a\mathrm{/}2{)}^2)\mathrm{/}(2\cdot v)=(8^2+(23\mathrm{/}2{)}^2)\mathrm{/}(2\cdot 8)={\displaystyle \frac{785}{64}}\doteq 12.2656 \\ V=\pi \cdot r^2\cdot d=3.1416\cdot 12.2656^2\cdot 24=11343.3{\text{dm}}^3 \end{gathered}$$

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