Final answer:
To find the radius of a new sphere formed by melting eight small spheres, we use the volume conservation method. The calculations reveal that the radius of the new sphere is 4 mm, which matches answer option (a).
Step-by-step explanation:
To calculate the radius of the new sphere formed by melting and casting eight metallic spheres, each with a radius of 2 mm, we need to conserve volume. The volume of a sphere is given by the formula V = \(\frac{4}{3}\pi r^3\), where V represents volume and r represents radius.
- Determine the volume of one of the small spheres: V_small = \(\frac{4}{3}\pi (2 mm)^3\).
- Calculate the total volume of all eight small spheres: Vtotal = 8 \times V_small.
- Set this total volume equal to the volume of the new larger sphere: Vtotal = \(\frac{4}{3}\pi R^3\), where R is the radius of the new sphere.
- Solve for R after substituting Vtotal into the equation.
After performing these calculations, the radius of the new sphere (R) is found to be 4 mm.