Question

What volume (in cc) is occupied by a block of wood of dimensions \(25.0 \mathrm{~m} \times 10.0 \mathrm{~cm} \times 300 \mathrm{~mm} .\) All edges are \(90^{\circ}\) to one another.

Short answer

The volume of the block of wood is 750,000.0 cubic centimeters (cc).

Step-by-step solution

Convert measurements to centimeters

We have the dimensions: Length = 25.0 m Width = 10.0 cm Height = 300 mm We will convert meters and millimeters to centimeters: 1 meter = 100 centimeters, so 25.0 m = 25.0 * 100 = 2500.0 cm 1 centimeter = 10 millimeters, so 300 mm = 300 / 10 = 30.0 cm Now the dimensions in centimeters are: Length = 2500.0 cm Width = 10.0 cm Height = 30.0 cm

Calculate the volume

To find the volume of a rectangular block, we multiply its length, width, and height: Volume = Length × Width × Height Substitute the dimensions in centimeters: Volume = 2500.0 cm × 10.0 cm × 30.0 cm Calculate the result: Volume = 750,000.0 cubic centimeters (cc) The volume of the block of wood is 750,000.0 cubic centimeters.

Key concepts

Unit Conversion

Unit conversion is crucial when solving mathematical problems, especially those involving measurements in different units. To simplify calculations, it's often beneficial to convert all measurements to the same unit system.
In our exercise, we began with a length in meters, width in centimeters, and height in millimeters. Here's how each conversion was made:

• Meters to Centimeters: Since there are 100 centimeters in a meter, convert meters by multiplying by 100. Thus, 25.0 meters becomes 2500.0 centimeters.
• Millimeters to Centimeters: There are 10 millimeters in a centimeter, so convert millimeters by dividing by 10. Hence, 300 millimeters turns into 30.0 centimeters.

By converting all dimensions to centimeters, we streamline our calculations and ensure accuracy in finding the volume.

Rectangular Prism

A rectangular prism, also known as a cuboid, is a three-dimensional shape with six rectangular faces. It's characterized by its length, width, and height, all of which intersect at 90-degree angles.
In this problem, we are dealing with a block of wood shaped as a rectangular prism. Here are some key characteristics:

• It has six faces, all of which are rectangles.
• The opposite faces are parallel and equal in size.
• All angles between the edges are right angles (90 degrees).

Understanding the basic properties of a rectangular prism can aid in comprehending more complex geometric concepts.

Geometry

Geometry is a field of mathematics that focuses on the properties and relationships of shapes, sizes, and relative positions of figures. In this exercise, geometric principles help us solve for the volume of a rectangular block.
Some essential concepts applied here include:

• Volume Calculation: The volume of a rectangular prism is calculated by multiplying its length, width, and height: \(\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\).
• Right Angles: The dimensions intersect at right angles, which defines the shape as a true rectangular prism.

Geometry not only aids in understanding how to perform these calculations but also provides a visual framework to conceptualize three-dimensional spaces.

Measurement in Centimeters

Using centimeters as a standard unit of measurement simplifies calculation and enhances consistency, especially in problems involving volume calculation.
When all measurements are expressed in the same unit, it's easier to multiply and calculate the results without the need for additional conversions. In this scenario, the block's dimensions were successfully converted to:

• Length = 2500.0 cm
• Width = 10.0 cm
• Height = 30.0 cm

This uniformity is key to ensuring that volume calculated is accurate: 750,000.0 cubic centimeters. Centimeters are particularly useful because they offer a manageable scale, allowing for precise measurements and calculations in many practical applications.