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Density is a fundamental concept that links mass and volume. It is defined as the mass of an object divided by its volume. The formula for density is: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] In our problem, density helps us understand how much mercury is needed to occupy a given volume. It's given that mercury has a density of 13.6 g/mL. This suggests that each milliliter of mercury has a mass of 13.6 grams. Understanding this relationship allows us to calculate how much space mercury takes up when we know its mass. By rearranging the formula to find volume, we have: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] This computation is crucial for converting the mass of mercury into volume, which is essential for determining further dimensions of the cylinder.

To solve many physics problems, converting mass into volume is a key step. With the mass of mercury being 105.5 grams and its density 13.6 g/mL, we can calculate the volume of mercury needed to fill the cylindrical tube. Using the formula: \[ V = \frac{\text{Mass}}{\text{Density}} = \frac{105.5}{13.6} \] After the calculations, the volume (\( V \)) is approximately 7.7574 cm³. This number tells us how much space the mercury actually occupies in the tube. It's equivalent to the total internal volume of the cylindrical tube.

Once we have the volume, calculating the radius of the cylinder becomes straightforward using the formula of cylinder volume: \[ V = \pi r^2 h \] Here, \( r \) is the radius, \( h \) is the height (or length in this problem), and \( V \) is the volume already known (approximately 7.7574 cm³). As the height is 12.7 cm, we rearrange the formula for the radius (\( r \)) as: \[ r^2 = \frac{V}{\pi h} \] Substituting the values gives: \[ r^2 = \frac{7.7574}{\pi \times 12.7} \] Solving this will provide \( r \), which is the measure we need for further calculations.

Once the radius (\( r \)) of the cylinder is determined from our calculations, finding the diameter is next. The formula for the diameter is twice the radius, given as: \[ d = 2r \] For a cylinder with a radius \( r \approx 0.698 \text{ cm} \), the diameter would be: \[ d \approx 2 \times 0.698 \approx 1.396 \text{ cm} \] This diameter gives us the tube's inner circle span from one side to the other. Understanding how to calculate the inner diameter is crucial because it's necessary for determining the capacity and ensuring mercury fills the tube completely.