Question

A bedroom has a volume of \(115 \mathrm{~m}^{3} .\) What is its volume in each unit? a. \(\mathrm{km}^{3}\) b. \(\mathrm{dm}^{3}\) c. \(\mathrm{cm}^{3}\)

Short answer

a. \(115 \mathrm{~m}^{3} = 1.15 \times 10^{-10} \mathrm{~km}^{3}\) b. \(115 \mathrm{~m}^{3} = 115 \times 10^{3} \mathrm{~dm}^{3}\) c. \(115 \mathrm{~m}^{3} = 115 \times 10^{6} \mathrm{~cm}^{3}\)

Step-by-step solution

Convert cubic meters to cubic kilometers

Use the conversion factor between cubic meters and cubic kilometers. There are 1,000,000,000,000 (or \(10^{12}\)) cubic meters in a cubic kilometer. Divide the volume in cubic meters by \(10^{12}\) to get the volume in cubic kilometers.

Convert cubic meters to cubic decimeters

Use the conversion factor between cubic meters and cubic decimeters. There are 1,000 cubic decimeters in a cubic meter. Multiply the volume in cubic meters by 1,000 to get the volume in cubic decimeters.

Convert cubic meters to cubic centimeters

Use the conversion factor between cubic meters and cubic centimeters. There are 1,000,000 cubic centimeters in a cubic meter. Multiply the volume in cubic meters by 1,000,000 to get the volume in cubic centimeters.

Key concepts

Unit Conversion

Unit conversion is a crucial skill in many scientific, technical, and everyday applications. It involves changing a measurement from one unit to another while maintaining the same quantity. Understanding unit conversion can help students solve problems across various subjects including physics, chemistry, and geometry. For example, in the textbook exercise, converting the volume of a bedroom from cubic meters to other units requires the use of specific conversion factors.

When attempting unit conversion, one must first understand the relationship between the units involved. The conversion factors used in the exercise are standard: one cubic kilometer contains \(10^{12}\) cubic meters; one cubic meter equals 1,000 cubic decimeters; and there are 1,000,000 cubic centimeters in one cubic meter. By using these relationships, students can confidently convert volumes as needed in a straightforward manner. Remember, in conversions, multiplying or dividing by the conversion factor is determined by the direction of change: dividing when going to a larger unit and multiplying when going to a smaller unit.

Cubic Meters

Cubic meters (\(m^3\)) are a unit of volume in the metric system, used to measure the space that an object occupies. Understanding the concept of cubic meters is essential for comprehending the scale of different volumes. For visualization, one cubic meter is the volume of a cube with edges that are each one meter in length. This measure is quite substantial in size and commonly used in contexts like specifying the volume of rooms or large containers.

When converting from cubic meters to another unit, it's beneficial to visualize how many of the smaller units fit inside one cubic meter. For instance, in the bedroom example, imagine how many smaller cubes of one cubic kilometer, cubic decimeter, or cubic centimeter can be packed into this space to complete the conversion.

Metric System

The metric system is a standardized system of measurement based on the number 10, which makes it exceptionally convenient for calculations and conversions. It's widely used around the world and includes units such as meters for length, liters for volume, and grams for mass. Volume measurements in the metric system typically include cubic meters, liters, and their derivatives like cubic centimeters and cubic decimeters.

In the exercise that required conversion of bedroom volume, each of the subsequent units—cubic kilometers (\(km^3\)), cubic decimeters (\(dm^3\)), and cubic centimeters (\(cm^3\))—are all part of this metric system. The importance of metric system consistency lies in its base-10 structure, enabling smooth transitions between units by simply moving the decimal point accordingly—a helpful attribute for students and professionals alike.