Cannonballs

Of the three cannonballs with a diameter of 16 cm, which landed in the castle courtyard during the battle, the castle blacksmith cast balls with a diameter of 10 cm, which fit into the cannons placed on the walls. How many cannonballs did the blacksmith cast?

Correct result:

n₂ =  12

Solution:

$$\begin{gathered} n_1=3 \\ {D}_1=16\text{ cm } \\ {D}_2=10\text{ cm } \\ {r}_1=D_1\mathrm{/}2=16\mathrm{/}2=8\text{ cm } \\ {r}_2=D_2\mathrm{/}2=10\mathrm{/}2=5\text{ cm } \\ {V}_1=n_1\cdot {\displaystyle \frac{4}{3}}\cdot \pi \cdot r_1^3=3\cdot {\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 8^3\doteq 6433.9818{\text{cm}}^3 \\ {V}_2={\displaystyle \frac{4}{3}}\cdot \pi \cdot r_2^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 5^3\doteq 523.5988{\text{cm}}^3 \\ {V}_1=n_2\cdot V_2 \\ {n}_2=\lfloor V_1\mathrm{/}{V}_2\rfloor =\lfloor 6433.9818\mathrm{/}523.5988\rfloor =12 \end{gathered}$$

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