Final answer:
To find the radius of a solid aluminum sphere that will balance a solid iron sphere on an equal-arm balance, we need to compare the mass and volume of the two spheres. By equating the volumes of the aluminum and iron spheres, we can find the radius of the aluminum sphere. The radius of the aluminum sphere that will balance the iron sphere is approximately 0.564 cm.
Step-by-step explanation:
To determine the radius of a solid aluminum sphere that will balance a solid iron sphere on an equal-arm balance, we need to consider the mass and volume of the two spheres.
The mass of 1 cubic meter (or 1,000,000 cubic centimeters) of aluminum is 2.70 x 10³ kg, and the mass of 1 cubic meter of iron is 7.86 x 10³ kg. Using these values, we can calculate the volume of the solid iron sphere.
Since the radius of the iron sphere is given as 2.00 cm, we can use the formula for the volume of a sphere (V = 4/3 * π * r^3) to calculate the volume of the iron sphere.
Once we have the volume of the iron sphere, we can equate it to the volume of the aluminum sphere using the mass-to-volume ratio of aluminum to find the radius of the aluminum sphere that will balance the iron sphere.
Using the known densities of aluminum and iron (2.70 g/cm³ and 7.86 g/cm³, respectively), we can convert the mass of the iron sphere to its volume and equate it to the volume of the aluminum sphere to find the radius.
By plugging in the values into the equations and solving, we find that the radius of the aluminum sphere that will balance the iron sphere is approximately 0.564 cm.