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?

Result

n2 =  12

Solution:

n1=3 D1=16 cm D2=10 cm  r1=D1/2=16/2=8 cm r2=D2/2=10/2=5 cm  V1=n1⋅ 43⋅ π⋅ r13=3⋅ 43⋅ 3.1416⋅ 83≐6433.9818 cm3  V2=43⋅ π⋅ r23=43⋅ 3.1416⋅ 53≐523.5988 cm3  V1=n2⋅ V2  n2=⌊V1/V2⌋=⌊6433.9818/523.5988⌋=12n_{1}=3 \ \\ D_{1}=16 \ \text{cm} \ \\ D_{2}=10 \ \text{cm} \ \\ \ \\ r_{1}=D_{1}/2=16/2=8 \ \text{cm} \ \\ r_{2}=D_{2}/2=10/2=5 \ \text{cm} \ \\ \ \\ V_{1}=n_{1} \cdot \ \dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{1}^3=3 \cdot \ \dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 8^3 \doteq 6433.9818 \ \text{cm}^3 \ \\ \ \\ V_{2}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{2}^3=\dfrac{ 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}/V_{2} \rfloor=\lfloor 6433.9818/523.5988 \rfloor=12

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