What volume (in cm') of lead fof density \(11.3 \mathrm{~g} \times \mathrm{cm}^{-1}\) ) has the same mass as \(100 \mathrm{~cm}^{\prime}\) of a piece of redwood (of density \(\left.0.38 \mathrm{~g} \mathrm{~cm}^{-1}\right)\) ?

Understanding the mass-volume relationship is critical when studying the characteristics of different materials. Imagine you have two objects, a sponge and a brick; while they might occupy a similar volume, their masses are quite different. This is because they have different densities, a concept we'll explore soon. In practical scenarios such as the textbook exercise, we compare the mass of redwood to that of lead by considering their volume and density. For a given material, the relationship between mass and volume is direct—if you increase the volume while keeping the density constant, the mass increases proportionally, and vice versa.

For scientific analyses and real-world applications, utilizing the mass-volume relationship allows us to replace materials while keeping certain properties consistent. For instance, the given exercise approaches this by aiming to match the mass of two materials with different densities by calculating and comparing their respective volumes.

The density of a substance, usually symbolized by the Greek letter ρ (rho), is a measure of its mass per unit volume. The formula to calculate density is quite straightforward: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]
In other words, if you want to find out how dense something is, you would divide the mass of the material by the volume it occupies. Higher density means more mass packed into a given volume.

This concept is crucial in many fields, from engineering to environmental science. When working through the given textbook problem, we apply the density formula inversely to find volume from the mass of the redwood and the density of lead. By rearranging the density formula, we get the volume formula: \[\text{Volume} = \frac{\text{Mass}}{\text{Density}}\]
The exercise demonstrates this application of the formula, reflecting the deeply intertwined nature of mass, volume, and density.

A key process in solving any scientific problem is ensuring that your units are consistent, which may require unit conversion. Within scientific calculations, converting units is as important as the calculations themselves. For example, if you have a volume in milliliters (mL) and density in grams per cubic centimeter (g/cm³), you'll need to convert mL to cm³ before using the density formula. This is because 1 mL is equal to 1 cm³, so ensuring that volume and density units match is imperative to avoid erroneous results.

In our example exercise, we fortunately don't need a conversion since both the mass of redwood and the density of lead are given in grams and cubic centimeters, respectively, which are compatible units. However, in different scenarios, not performing the necessary unit conversions can lead to mistakes. Always double-check that your units align for accurate calculations in density or any other formula-driven determination.