The density of lead is 11.34 g/cm3 and the density of aluminum is 2.69 g/cm3. Find the radius of lead and aluminum spheres each having a mass of 10 kg. (Round your answers to two decimal places.)

To find the radius of lead and aluminum spheres each having a mass of 10 kg, we can follow these steps:

### Step 1: Use the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.

### Step 2: Use the relationship between mass, density, and volume
The mass \( m \) of an object is related to its density \( \rho \) and volume \( V \) by the equation:
\[
m = \rho V
\]
We can rearrange this equation to find the volume:
\[
V = \frac{m}{\rho}
\]

### Step 3: Calculate the volume of the lead sphere
Given:
- Mass of lead sphere \( m_{\text{lead}} = 10 \, \text{kg} = 10,000 \, \text{g} \)
- Density of lead \( \rho_{\text{lead}} = 11.34 \, \text{g/cm}^3 \)

The volume of the lead sphere is:
\[
V_{\text{lead}} = \frac{10,000 \, \text{g}}{11.34 \, \text{g/cm}^3}
\]
Let's calculate the volume.

### Step 4: Calculate the radius of the lead sphere
Once we have the volume, we can use the formula for the volume of a sphere:
\[
r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}}
\]
We'll calculate the radius after determining the volume.

### Step 5: Calculate the volume of the aluminum sphere
Given:
- Mass of aluminum sphere \( m_{\text{aluminum}} = 10 \, \text{kg} = 10,000 \, \text{g} \)
- Density of aluminum \( \rho_{\text{aluminum}} = 2.69 \, \text{g/cm}^3 \)

The volume of the aluminum sphere is:
\[
V_{\text{aluminum}} = \frac{10,000 \, \text{g}}{2.69 \, \text{g/cm}^3}
\]
Let's calculate the volume.

### Step 6: Calculate the radius of the aluminum sphere
Once we have the volume, we can use the same formula to find the radius of the aluminum sphere:
\[
r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}}
\]

Now, let's calculate both radii.The radius of the spheres are as follows:

- Lead sphere: \(5.95 \, \text{cm}\)
- Aluminum sphere: \(9.61 \, \text{cm}\)

These values are rounded to two decimal places.

Would you like further details on any step or have any questions?

Here are 5 related questions for further exploration:
1. How does changing the mass affect the radius of a sphere?
2. What would be the radius of a copper sphere with a mass of 10 kg?
3. How does the density of a material impact the size of objects with the same mass?
4. What is the relationship between volume and radius in a sphere?
5. How would you calculate the surface area of these spheres?

**Tip:** The density of a material is crucial in determining how much space a given mass will occupy. Higher density materials will form smaller objects for the same mass.

Learn how to calculate the radius of lead and aluminum spheres, each with a mass of 10 kg. This problem involves key geometric concepts like the volume of spheres, density, and algebra. Understand how density affects object size, with step-by-step solutions. Suitable for students in Grades 10-12.

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