Answer:
Step-by-step explanation:
Recall that the volume of a sphere is equal to [tex]\frac{4}{3} \pi r^{3}[/tex]. First, let's find the volume of 1 of these spheres:
[plug in value]
[tex]\frac{4}{3} \pi (2mm)^{3}[/tex]
[simplify]
[tex]\frac{32\pi}{3} mm^{3}[/tex] [exact]
or
33.51 mm³ [approximate]
We are not done yet because we have 8 of these spheres, not just one. Luckily for us, they are all the same dimensions, so we can multiply the area found by 8:
33.51 mm³ * (8)
268.1 mm³, which is the final answer.
Answer:
8 mm
Step-by-step explanation:
To find the radius of the new sphere formed by melting and casting the 8 metallic spheres, we can apply the principle of conservation of volume, which states that the total volume remains constant.
The formula for the volume of a sphere is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Sphere}}\\\\V=\dfrac{4}{3}\pi r^3\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\end{array}}[/tex]
Therefore, the total volume of the 8 metallic spheres, each with radius of 2 mm, is:
[tex]V=8 \cdot \dfrac{4}{3} \pi \cdot 2^3\\\\\\V=8 \cdot \dfrac{4}{3} \pi \cdot 8\\\\\\V=\dfrac{256}{3} \pi[/tex]
When these 8 spheres are melted and cast into a single sphere, the volume remains the same. Therefore, to find the radius of the new sphere, set the volume of a sphere formula equal to 256π/3 mm³ and solve for r:
[tex]\dfrac{4}{3}\pi r^3=\dfrac{256}{3}\pi\\\\\\\dfrac{4}{3} r^3=\dfrac{256}{3}\\\\\\4r^3=256\\\\\\r^3=\dfrac{256}{4}\\\\\\r^3=64\\\\\\r=\sqrt[3]{64}\\\\\\r=8\; \sf mm[/tex]
Therefore, the radius of the new sphere is:
[tex]\LARGE\boxed{\boxed{8\; \sf mm}}[/tex]