Problem 162 - Introductory Chemistry Atoms First

Question

A block measures \(6.0 \mathrm{~cm}\) on each side. What is the volume of the block in cubic meters?

Step-by-Step Solution

Answer

The volume of the block is approximately \(0.000216 m^3\).

Step 1: Identify the formula for the volume of a cube

Since the block is a cube, we can calculate the volume using the formula: Volume = side^3 where side is the length of each side of the cube.

Step 2: Calculate the volume in cubic centimeters

Given that each side of the cube measures 6.0 cm, we can find the volume as follows: Volume = (6.0 cm)^3 = 216.0 cm^3

Step 3: Convert the volume from cubic centimeters to cubic meters

We know that 1 m = 100 cm, so 1 m^3 = (100 cm)^3 = 1,000,000 cm^3. To convert the volume from cubic centimeters to cubic meters, we can use the following formula: Volume (m^3) = Volume (cm^3) / 1,000,000

Step 4: Calculate the volume in cubic meters

Now we can find the volume in cubic meters using the conversion formula: Volume (m^3) = 216.0 cm^3 / 1,000,000 = 0.000216 m^3 So, the volume of the block is approximately 0.000216 cubic meters.

Key Concepts

Cubic Meters Conversion

Converting units of volume can sometimes feel tricky, but it's a straightforward process once you have the key formula in hand. The key is understanding the relationship between cubic centimeters and cubic meters.

One cubic meter ( 1 ext{m}^3 ) is a big space compared to a cubic centimeter ( 1 ext{cm}^3 ). In fact, it takes one million cubic centimeters to fill one cubic meter. This is because the conversion relies on the cube of the linear conversion factor (100 cm = 1 m), leading us to 1 ext{m}^3 = 100^3 ext{cm}^3 = 1,000,000 ext{cm}^3 .

To convert any volume from cubic centimeters to cubic meters, simply use the conversion formula:

• Volume in meters ( ext{m}^3 ) = Volume in centimeters ( ext{cm}^3 ) / 1,000,000

This formula helps ensure that you're converting accurately and effectively.

Measurement Units

Measurement units are essential in everyday life, especially when dealing with geometric figures. Understanding units is crucial as each unit measures different quantities, like length, area, and volume. Here we're focused on volume

When measuring volume, common units include:

• Cubic meters ( ext{m}^3 )
• Cubic centimeters ( ext{cm}^3 )
• Liters ( 1 ext{L} = 1,000 ext{cm}^3 )

Each unit serves a specific purpose based on the quantity's size being measured. Cubic meters are used for large volumes like pools, while cubic centimeters fit smaller objects like ice cubes better.

Knowing different measurement units and how to convert between them is key in interpreting results accurately!

Cubic Centimeters

Cubic centimeters are a fundamental unit of volume in the metric system. They represent a cube with each side measuring one centimeter. This makes them great for smaller objects where precise measurements are needed.

To imagine, if you think of a small sugar cube, it's roughly 1 ext{cm}^3 in volume!

When calculating volume using length measurements in centimeters, the resulting volume is naturally expressed in cubic centimeters. This is important in geometry exercises when measuring the volume of objects whose dimensions are given in centimeters.

Always remember, any calculation involving volume and conversions may require returning to the cubic centimeter as a base unit before converting to larger or smaller units. This ensures maximum precision.

Geometry Formulas

Understanding geometry formulas is critical to solving volume-related problems. These formulas provide a structured way to calculate dimensions and volumes of different objects, like cubes, cylinders, etc., using their specific properties.

For a cube, the volume formula is particularly simple and often memorized:

• Volume = (side)^3

This formula states that the cube's volume is the cube of the length of its side. Having this formula at your fingertips empowers you to easily calculate the volume of any cubic object once you have the side length.

These geometry formulas are powerful tools, whether for practical applications or academic exercises, as they help break complex concepts into manageable calculations.