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The problem displayed in the image involves determining the density and radius of a ball bearing using measurements given.

### Problem Analysis and Calculation:

1. **Finding the density of the ball bearing**:
   The difference in the mass of the cylinder before and after the addition of the mineral oil gives the mass of the oil. Subtracting the combined mass of the ball bearing and mineral oil by the mass of the oil gives the mass of the ball bearing alone.

   \[
   \text{Mass of the oil} = 1549.46 \text{ g} - 1249.66 \text{ g} = 299.8 \text{ g}
   \]
   \[
   \text{Mass of the ball bearing} = 1509.52 \text{ g} - 299.8 \text{ g} = 1209.72 \text{ g}
   \]

   Density (\(\rho\)) is mass per unit volume. The volume of the ball bearing can be calculated using the displacement method (since the cylinder with the ball bearing was filled back to 40.00 mL). Assuming the original 40.00 mL is the total volume of oil plus ball bearing, the volume of the ball bearing is the total volume minus the volume displaced by the oil:

   \[
   \text{Volume of ball bearing} = 40.00 \text{ mL} - (40.00 \text{ mL} - \text{Volume displaced by oil}) = \text{Volume displaced by oil}
   \]

   Since the density of the mineral oil is not given directly, and we don't know how much volume the oil displaced, we assume that the volume of the oil equals the volume of the ball bearing due to displacement:

   \[
   \text{Density of the ball bearing} = \frac{1209.72 \text{ g}}{40.00 \text{ mL}} = 30.243 \text{ g/mL}
   \]

2. **Finding the radius of the ball bearing**:
   Assuming the ball bearing is spherical, we use the formula for the volume of a sphere:

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

   Solving for \(r\):

   \[
   r = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}
   \]

   Substituting \(V = 40.00 \text{ mL}\):

   \[
   r = \left(\frac{3 \times 40.00}{4\pi}\right)^{\frac{1}{3}}
   \]

Calculating the radius using this formula:

```python
import math

# Constants
volume = 40.00  # in mL

# Calculate the radius in cm (1 mL = 1 cm³)
radius = ((3 * volume) / (4 * math.pi)) ** (1/3)

radius
```The calculations yield the following results:

1. **Density of the ball bearing**: \(30.243 \, \text{g/mL}\) (as computed manually above).
2. **Radius of the ball bearing**: approximately \(2.12 \, \text{cm}\).

These values can be entered into the respective fields on the website.

Would you like more details on any part of this solution or have any other questions?

### Follow-up Questions:
1. Would you like to explore how changes in the volume or mass would affect the density?
2. Are you interested in learning about error analysis for this type of experiment?
3. Would you like to calculate the surface area of the ball bearing given its radius?
4. Do you need to know how to convert the density into different units, like kg/m³?
5. Are you interested in the properties of materials that might have similar densities?

### Tip:
Remember, always ensure to use significant figures correctly based on the input data in scientific calculations.

A graduated cylinder is filled to the 40.00 mL mark with a mineral oil. The masses of the cylinder before and after the addition of the mineral oil are 1249.66 g and 1549.46 g, respectively. In a separate experiment, a metal ball bearing of mass 21.431 g is placed in the cylinder and the cylinder is again filled to the 40.00 mL mark with the mineral oil. The combined mass of the ball bearing and mineral oil is 1509.52 g. Calculate the density and radius of the ball bearing.

Learn how to calculate the density and radius of a ball bearing using mass and volume displacement. This problem involves key physics and geometry concepts such as density calculation and the volume of a sphere. Suitable for advanced high school students.

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