Question

A classroom has a volume of \(285 \mathrm{~m}^{3}\). What is its volume in each unit? (a) \(\mathrm{km}^{3}\) (b) \(\mathrm{dm}^{3}\) (c) \(\mathrm{cm}^{3}\)

Short answer

The volume of the classroom is (a) \( 2.85 \times 10^{-7} \text{km}^3 \), (b) \( 285 \times 10^3 \text{dm}^3 \), (c) \( 285 \times 10^6 \text{cm}^3 \).

Step-by-step solution

Convert cubic meters to cubic kilometers

To convert cubic meters to cubic kilometers, recall that there are \(10^3\) meters in a kilometer. Thus, there are \(10^3\)^3 or \(10^9\) cubic meters in a cubic kilometer. Use the conversion factor \(1 \text{km}^3 = 10^9 \text{m}^3\). Divide the volume in cubic meters by \(10^9\).

Convert cubic meters to cubic decimeters

Knowing that there are 10 decimeters in a meter, then there are \(10^3\) cubic decimeters in a cubic meter. Therefore, multiply the volume in cubic meters by \(10^3\).

Convert cubic meters to cubic centimeters

Since there are \(10^2\) centimeters in a meter, there are \(10^2\)^3 or \(10^6\) cubic centimeters in a cubic meter. Multiply the volume in cubic meters by \(10^6\) to convert to cubic centimeters.

Key concepts

Unit Conversion

Unit conversion is a multi-step process used to change the units of a given measurement without changing the value of the measurement. This is a critical skill in various fields such as science, engineering, and everyday life when working with different measurement systems. To facilitate conversions, we often use conversion factors which are ratios that represent the relationship between different units.

For example, when converting from one square unit to another, like square meters to square kilometers, we use the fact that 1 km equals 1,000 meters. This relationship helps us understand that the conversion factor will have a large impact on our result, often involving raising the conversion factor to the power that represents the dimension – like the cubic power for volume conversions.

It's vital to keep in mind that converting units can heavily affect numerical values, making them either significantly larger or smaller, which can be crucial when making calculations in different contexts.

Cubic Meters

Cubic meters, represented as \( m^3 \), are a unit of volume commonly used in the International System of Units (SI). It's defined as the volume of a cube with edges that are one meter in length. Understanding cubic meters is essential because it helps in visualizing the amount of three-dimensional space an object takes up.

For example, the volume of a classroom that measures \(285 m^3\) represents a substantial amount of space. When we look at room dimensions or water in a swimming pool, we're often talking about cubic meters. It is pivotal to comprehend this unit because it is a fundamental measurement within numerous fields such as construction, architecture, and in environmental studies when discussing air quality or water reserves.

Dimensional Analysis

Dimensional analysis is a technique used to convert from one set of units to another. This approach involves using conversion factors to cancel out units until you are left with the desired unit. It is also widely known for its role in ensuring that equations are consistent in terms of their units.

When carrying out dimensional analysis, experts often create a train of conversion factors and align units diagonally in such a way that undesired units cancel out to give the preferred units. This ensures the accuracy of calculations, especially in complex scenarios involving multiple unit conversions.

This method is particularly important in scientific measurements where precision is paramount, and a misunderstanding or miscalculation can lead to significant errors with serious consequences. Dimensional analysis serves as a powerful tool to guide through the conversion maze and ensure that the results are logically consistent and practically applicable.