Question

A piece of black onyx was cut into a rectangular solid measuring \(5.00 \mathrm{~cm}\) by \(5.00 \mathrm{~cm}\) by \(2.50 \mathrm{~mm}\). What is the volume in cubic centimeters?

Short answer

The volume is 6.25 cubic centimeters.

Step-by-step solution

Convert Millimeters to Centimeters

Since the thickness is given in millimeters, first we need to convert it to centimeters. We note that 10 millimeters is equivalent to 1 centimeter. Therefore, \[2.50 \text{ mm} = \frac{2.50}{10} \text{ cm} = 0.25 \text{ cm}.\]

Calculate Volume using Formula

The volume of a rectangular solid is given by the formula \( \text{Volume} = \text{length} \times \text{width} \times \text{height} \). So, substituting the values,\[\text{Volume} = 5.00 \text{ cm} \times 5.00 \text{ cm} \times 0.25 \text{ cm} = 6.25 \text{ cm}^3.\]

Key concepts

Unit Conversion

In the realm of volume calculation, converting units can be crucial, especially when different units are involved in a single problem. In our exercise, we encounter measurements given in centimeters and millimeters. To ensure uniformity, millimeters were converted to centimeters, which is the unit for all other dimensions.

Unit conversion, especially from millimeters to centimeters, follows a simple rule:

• 1 centimeter is equal to 10 millimeters.
• Therefore, to convert millimeters to centimeters, you divide the millimeters by 10.

For example, converting 2.50 millimeters into centimeters involves dividing by 10, giving us 0.25 centimeters. This step is essential as it consolidates units, allowing for straightforward calculations of volume without complications arising from mixed units.

Rectangular Solid

A rectangular solid, also known as a rectangular prism, is a three-dimensional object where each face is a rectangle. It is a common shape for objects, making its volume determination a central skill in geometry.

The key aspects of a rectangular solid include:

• Length: The longest side or edge of the rectangle.
• Width: The shorter side on the same plane as the length.
• Height: The distance from the base of the rectangle to its top.
• All angles are right angles, and opposite faces are equal.

To find the volume of a rectangular solid, you multiply its length, width, and height together. This results in a measure of how much space the solid occupies, expressed in cubic units. For an object measuring 5.00 cm by 5.00 cm by 0.25 cm, the volume is simply calculated, giving a clear understanding of spatial occupation in three dimensions.

Measurement in Centimeters

Using centimeters in volume calculations is common due to their convenience and prevalence in metric measurements. When all dimensions are in centimeters, calculations become straightforward and avoid potential errors arising from unit mismatches.

Here's why centimeters are often preferred:

• It is part of the metric system, which is widely used in scientific and everyday measurement.
• Centimeters offer a balance between precision and manageability, unlike millimeters (too small) or meters (too large) for everyday objects.
• The ease of converting related metric units, such as from millimeters or meters, facilitates straightforward calculations.

In the given exercise, after ensuring all dimensions were in centimeters, the calculation of volume proved to be more efficient. This consistency underscores the importance of selecting appropriate units for problem-solving.