The cylindrical container

The cylindrical container has a base area of 300 cm³ and a height of 10 cm. It is 90% filled with water. We gradually insert metal balls into the water, each with a volume of 20 cm³. After inserting how many balls for the first time does water flow over the edge of the container?

Correct result:

n =  16

Solution:

$$\begin{gathered} S=300{\text{cm}}^2 \\ h=10\text{ cm } \\ k=90\mathrm{\%}={\displaystyle \frac{90}{100}}=0.9 \\ {h}_1=k\cdot h=0.9\cdot 10=9\text{ cm } \\ {V}_0=S\cdot h=300\cdot 10=3000{\text{cm}}^3 \\ V=k\cdot V_0=0.9\cdot 3000=2700{\text{cm}}^3 \\ {V}_1=20{\text{cm}}^3 \\ {V}_1={\displaystyle \frac{4}{3}}\pi r^3 \\ r=\sqrt[3]{3\cdot V_1\mathrm{/}4\mathrm{/}\pi }=\sqrt[3]{3\cdot 20\mathrm{/}4\mathrm{/}3.1416}\doteq 1.6839\text{ cm } \\ D=2\cdot r=2\cdot 1.6839\doteq 3.3678\text{ cm } \\ D<h_1 \\ {n}_1=(V_0-V)\mathrm{/}{V}_1=(3000-2700)\mathrm{/}20=15 \\ n=\lceil n_1\rceil +1=\lceil 15\rceil +1=16 \end{gathered}$$

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