One cubic meter 1.00 m3 of aluminum has a mass of 2.70E+3 kg, and 1.00 m3 of copper has a mass of 8.92E+3 kg. Find the radius of an aluminum sphere whose mass is the same as that of an copper sphere of radius 1.85 cm

To solve the problem, we need to find the radius of an aluminum sphere whose mass is equal to the mass of a copper sphere with a radius of 1.85 cm.

### Step 1: Find the mass of the copper sphere
The mass \( m \) of a sphere is given by the formula:

\[
m = \rho \cdot V
\]

where:
- \( \rho \) is the density of the material,
- \( V \) is the volume of the sphere.

The volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

For copper:
- \( \rho_{\text{Cu}} = 8.92 \times 10^3 \, \text{kg/m}^3 \) (density of copper),
- \( r_{\text{Cu}} = 1.85 \, \text{cm} = 0.0185 \, \text{m} \).

Now, calculate the volume of the copper sphere:

\[
V_{\text{Cu}} = \frac{4}{3} \pi (0.0185)^3
\]

Substitute this into the mass equation for copper:

\[
m_{\text{Cu}} = 8.92 \times 10^3 \times \frac{4}{3} \pi (0.0185)^3
\]

### Step 2: Set up the equation for the aluminum sphere
The mass of the aluminum sphere must equal the mass of the copper sphere:

\[
m_{\text{Al}} = m_{\text{Cu}}
\]

For aluminum:
- \( \rho_{\text{Al}} = 2.70 \times 10^3 \, \text{kg/m}^3 \),
- The volume of the aluminum sphere is \( V_{\text{Al}} = \frac{4}{3} \pi r_{\text{Al}}^3 \).

The mass of the aluminum sphere is:

\[
m_{\text{Al}} = 2.70 \times 10^3 \times \frac{4}{3} \pi r_{\text{Al}}^3
\]

Equating the masses of the copper and aluminum spheres:

\[
8.92 \times 10^3 \times \frac{4}{3} \pi (0.0185)^3 = 2.70 \times 10^3 \times \frac{4}{3} \pi r_{\text{Al}}^3
\]

### Step 3: Solve for \( r_{\text{Al}} \)
The \( \frac{4}{3} \pi \) terms cancel out, leaving:

\[
8.92 \times (0.0185)^3 = 2.70 \times r_{\text{Al}}^3
\]

Now solve for \( r_{\text{Al}}^3 \):

\[
r_{\text{Al}}^3 = \frac{8.92 \times (0.0185)^3}{2.70}
\]

Finally, take the cube root of both sides to find \( r_{\text{Al}} \).

Let me calculate this for you.The radius of the aluminum sphere, whose mass is the same as that of a copper sphere with a radius of 1.85 cm, is approximately \( 0.0276 \, \text{m} \), or 2.76 cm.

Would you like further details or clarification on any steps?

Here are some related questions to expand on this concept:
1. How would the radius change if the copper sphere had a different density?
2. What would be the mass of an aluminum sphere with a radius of 1.85 cm?
3. How would the calculation change if the shapes were cylinders instead of spheres?
4. What is the volume difference between the aluminum and copper spheres for the same mass?
5. How would the result change if the copper sphere was hollow instead of solid?

**Tip:** In problems involving different materials with equal masses, the ratio of their volumes is inversely proportional to their densities.

One cubic meter 1.00 m3 of aluminum has a mass of 2.70E+3 kg, and 1.00 m3 of copper has a mass of 8.92E+3 kg. Find the radius of an aluminum sphere whose mass is the same as that of a copper sphere of radius 1.85 cm.

Discover how to find the radius of an aluminum sphere that has the same mass as a copper sphere with a 1.85 cm radius. This problem involves calculating volume, understanding the density-mass relationship, and using key geometry formulas. Suitable for students in Grades 10-12.

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