Final answer:
When a sphere's dimensions are scaled by a factor of 200, its volume increases by a factor of 8,000,000, since volume scales with the cube of the scaling factor (200³ = 8,000,000).
Step-by-step explanation:
If a sphere is proportionally scaled by a factor of 200, the effect on its volume will be significant due to the properties of geometric scaling. The volume of a sphere scales with the cube of the radius. So, when the scaling factor is 200, we must cube this factor to determine the change in volume.
The original volume of the sphere can be given by the formula V = (4/3)πr³, where r is the radius of the sphere. If we scale the radius by a factor of 200, the new radius becomes 200r. To find the new volume V', we use the new radius in the volume formula:
V' = (4/3)π(200r)³
Therefore, V' equals (4/3)π200³r³, which simplifies to (4/3)π8,000,000r³. Since 200³ equals 8,000,000, we can conclude that the volume will increase by a factor of 8,000,000, which is much larger than any of the options provided in the original question. This means the correct answer, which isn't listed in the options, should indeed be that the volume of the sphere increases by a factor of 8,000,000 when the sphere is proportionally scaled by 200.