A cylindrical glass tube \(12.7 \mathrm{~cm}\) in length is filled with mercury. The mass of mercury needed to fill the tube is found to be \(105.5 \mathrm{~g}\). Calculate the inner diameter of the tube. (The density of mercury \(=13.6 \mathrm{~g} / \mathrm{mL}\).)

Understanding the density formula is essential when it comes to calculating the mass or volume of a substance. Density is defined as the mass of an object divided by its volume, which can be written in a mathematical formula as:

\( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

For the case of mercury in our cylindrical glass tube, knowing the density allows you to work out how much space a certain mass of mercury will occupy. Since density is a constant for a given substance at a specific temperature and pressure, it can be really useful for identifying substances or determining their quantities. For students trying to determine a missing measurement in exercises, grasping the density formula means that if two out of the three variables (mass, density, volume) are known, the third can be calculated. In the example, by rearranging the formula to solve for volume:

\( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)

you can find out how much space the mercury takes up inside the tube.

The volume of mercury, or any liquid, is the amount of space that it occupies. In our example, the volume of the mercury that fills the cylindrical tube is calculated using the rearranged density formula where mass is divided by density. With a mass of 105.5 grams and a known density of 13.6 g/mL, the volume is:

\( \text{Volume of Mercury} = \frac{105.5 \text{ g}}{13.6 \text{ g/mL}} = 7.75 \text{ mL} \)

Understanding the concept of liquid volume is crucial not only for the sake of this problem but also in various scientific and engineering applications, where precise measurements of liquids are often required. It's also worth noting that for liquids like mercury that don't compress under normal conditions, their volume is consistent regardless of the shape of the container they're in—which is why we can equate the volume of mercury to the volume of the cylinder it fills.

A cylinder's volume is the amount of space within the 3-dimensional shape, and it's calculated using the formula:

\( V = \pi r^2 h \)

where \(V\) is the volume, \(r\) is the radius of the cylinder's base circle, \(h\) is the height of the cylinder, and \(\pi\) is approximately equal to 3.14159. By knowing either the volume and height or the volume and radius, you can calculate the remaining dimension. This is what allows us to determine the inner diameter of our cylindrical mercury-filled tube. Since we are given the height and have calculated the volume of the mercury (which is equivalent to the volume of the cylinder), we can rearrange the formula to solve for the radius. Once we have the radius, it’s only a small step forward to find the cylinder’s diameter, which is required for the problem.

To calculate the diameter of any cylinder, we first need to know the radius. The diameter is simply twice the length of the radius, expressed in the equation:

\( d = 2r \)

After determining the volume of the cylinder using the mercury volume and the cylinder volume formula, solving for the radius, and then applying this simple multiplication, we can solve for the diameter of the cylindrical tube in our problem. Remember, the diameter runs from edge to edge of the circle, passing through the center. It's important for students to appreciate the relationship between radius, diameter, and circumference for full understanding of circular shapes and the measurements related to them. Calculating diameter is a fundamental skill that comes into play in a wide range of scientific and mathematical problems.