An iceberg has a volume of \(8975 \mathrm{ft}^{3}\). What is the mass in kilograms of the iceberg? The density of ice is \(0.917 \mathrm{~g} / \mathrm{cm}^{3}\).

One of the fundamental skills in scientific and mathematical problem solving is unit conversion. This is the process of converting a measure in one unit to its equivalent value in another unit, using a conversion factor. In the given exercise, the volume of an iceberg is originally provided in cubic feet, yet the density of ice is given in grams per cubic centimeter. To accurately compute mass, these units must be consistent.

To convert cubic feet to cubic centimeters, apply the conversion factor, where one foot is equivalent to 30.48 centimeters. Mathematically, this conversion requires you to cube the conversion factor since volume is a three-dimensional measure. For instance, if you have a volume of 1 cubic foot, you'd multiply it by \(30.48\ cm/ft\)^3 to find the equivalent in cubic centimeters. This step is crucial to proceed with mass calculations and ensures that your final result will be accurate and meaningful.

Understanding the mass-volume relationship is key when dealing with physical sciences and engineering concepts. Mass refers to the amount of matter in an object, while volume measures the space that object occupies. The density of a material can be thought of as its 'compactness', or how much mass is contained in a given volume.

In practice, if you know the volume of a substance and its density, you can calculate its mass by multiplying the two. This relationship is commonly written as \(mass = density \times volume\). In the exercise, once you've converted the volume to a consistent unit (cubic centimeters), multiplying it by the density of ice (given in grams per cubic centimeter) provides the mass of the iceberg in grams. Remembering the fundamental formula for density can facilitate a range of scientific calculations and applications.

The density of ice plays a pivotal role in this problem, and it reveals much about the properties of ice itself. Density is defined as mass per unit volume, and for ice, it generally has a value of approximately \(0.917 \ g/cm^3\). This value is less than the density of liquid water, which is why ice floats.

When you're given the density of any substance, it serves as a conversion factor between mass and volume for that specific substance. This characteristic allows one to calculate how much space (volume) a certain mass will occupy, or conversely, how heavy (mass) a known volume will be. For our iceberg, this density indicates that every cubic centimeter of ice has a mass of 0.917 grams, establishing the relationship essential for solving the mass from the given volume.

Effective scientific problem-solving requires a systematic approach to addressing questions and finding solutions. Students can practice this skill set by breaking down complex problems into smaller, more manageable steps. In our example, the process begins with understanding the problem, which includes identifying known information (volume and density) and what is being sought (mass).

Next, the application of unit conversion ensures compatibility of measurements. The subsequent course of action, using the mass-volume relationship to solve for mass, involves applying a formula or principle based on the known variables. Finally, converting the resulting mass into the desired units completes the process. This structured approach not only helps solve the problem at hand but also instills a logical methodology that can be applied across diverse scientific challenges.