A cylinder has a circular cross section with a radius of \(2.500 \mathrm{~cm}\). What is the volume of water in the cylinder with a measured height of \(1.20 \mathrm{~cm}\) ? (The volume of a cylinder is \(\pi r^{2} h\), where \(r\) is the radius and \(h\) is the height.)

Geometric formulas are essential tools for solving problems involving shapes and spaces. These formulas allow us to calculate various properties of geometric figures, such as area, volume, and surface area. In our case, working with a cylinder, we need to understand how its dimensions relate to its volume. A basic geometric principle for cylinders tells us that the volume is directly related to two main components: the area of its circular base and its height.

Understanding geometric formulas is not only instrumental in mathematics but also in practical applications, such as in constructing everyday objects and in scientific measurements. Thus, having a solid grasp of these formulas means we can navigate real-world scenarios effectively.

Calculating the volume of a cylinder is relatively straightforward. The formula provided V = \(\pi r^2 h\), where \(V\) represents volume, \(r\) radius, and \(h\) height, expresses this mathematical relationship. To compute the volume, one must first square the radius of the cylinder's base, multiply it by \(\pi\), the constant representing the ratio of a circle's circumference to its diameter, and then multiply this product by the cylinder's height.

Visualization of the Calculation

Imagine piling up a series of circles (the base of the cylinder) equal to the height of the cylinder. By doing this, you can visualize how the area of the base and the height together determine the total volume of the cylinder. The visualization can serve as a helpful mnemonic device for remembering this formula—essentially, the volume of a cylinder is just the area of the base multiplied by the height.

Mathematics is a foundational component in the field of chemistry, particularly when dealing with measurements and quantities of substances. Chemists frequently use mathematical concepts, like the volume of a cylinder, when working with liquids in laboratory settings. Knowing the precise volume of substances is vital for conducting experiments, as many chemical reactions depend on accurate measurements of reactants.

In our exercise, the cylinder could represent a graduated cylinder used in a chemistry lab to measure the volume of a liquid. By applying the cylinder volume calculation, we can accurately determine how much of a chemical is present or required for a reaction. This reinforces the importance of mathematical precision in scientific inquiry, underpinning how deeply interconnected mathematics is with the empirical sciences such as chemistry.