To solve the problem, we need to find the radius of an aluminum sphere with the same mass as that of a copper sphere with a known radius of 1.85 cm.

Step-by-Step Solution:

1. Find the mass of the copper sphere:

   • The volume $V$ of a sphere is given by: $V = \frac{4}{3} \pi r^3$
   • The density of copper is given as $\rho_{\text{Cu}} = 8.92 \times 10^3 \, \text{kg/m}^3$.
   • The radius $r_{\text{Cu}}$ of the copper sphere is 1.85 cm = 0.0185 m.
   • The mass of the copper sphere $m_{\text{Cu}}$ is: $m_{\text{Cu}} = \rho_{\text{Cu}} \times V_{\text{Cu}} = \rho_{\text{Cu}} \times \left( \frac{4}{3} \pi r_{\text{Cu}}^3 \right)$ Substituting the values, we get: $m_{\text{Cu}} = 8.92 \times 10^3 \times \frac{4}{3} \pi (0.0185)^3$

2. Set the mass of the aluminum sphere equal to the mass of the copper sphere:

   • Let $r_{\text{Al}}$ be the radius of the aluminum sphere.
   • The density of aluminum is $\rho_{\text{Al}} = 2.70 \times 10^3 \, \text{kg/m}^3$.
   • The mass of the aluminum sphere $m_{\text{Al}}$ is: $m_{\text{Al}} = \rho_{\text{Al}} \times \left( \frac{4}{3} \pi r_{\text{Al}}^3 \right)$
   • Since $m_{\text{Al}} = m_{\text{Cu}}$, we equate the masses: $\rho_{\text{Al}} \times \left( \frac{4}{3} \pi r_{\text{Al}}^3 \right) = \rho_{\text{Cu}} \times \left( \frac{4}{3} \pi r_{\text{Cu}}^3 \right)$

3. Solve for $r_{\text{Al}}$:

   • Simplifying the equation: $\rho_{\text{Al}} r_{\text{Al}}^3 = \rho_{\text{Cu}} r_{\text{Cu}}^3$ $r_{\text{Al}}^3 = \frac{\rho_{\text{Cu}}}{\rho_{\text{Al}}} r_{\text{Cu}}^3$ $r_{\text{Al}} = \left( \frac{\rho_{\text{Cu}}}{\rho_{\text{Al}}} \right)^{1/3} r_{\text{Cu}}$
   • Substituting the values: $r_{\text{Al}} = \left( \frac{8.92 \times 10^3}{2.70 \times 10^3} \right)^{1/3} \times 0.0185$

Let's calculate the final value.The radius of the aluminum sphere is approximately $0.0276 \, \text{m}$ or 2.76 cm.

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. How does the mass of a sphere change with its radius?
2. What is the formula for the volume of a sphere?
3. How do different materials' densities affect the size of objects with the same mass?
4. What is the relationship between mass, volume, and density?
5. How do you convert between cubic meters and cubic centimeters?

Tip: When comparing objects of the same mass but different materials, remember that the one with lower density will have a larger volume.