Problem 56 - College Physics

Question

A cord of wood is a volume of cut wood equal to a stack \(8.0 \mathrm{ft}\) long, \(4.0 \mathrm{ft}\) wide, and \(4.0 \mathrm{ft}\) high. How many cords are there in \(3.0 \mathrm{~m}^{3}\) ?

Step-by-Step Solution

Answer

There are approximately 0.827 cords in 3.0 cubic meters.

Step 1: Convert Cord Dimensions from Feet to Cubic Feet

The volume of a cord of wood is calculated as the product of its length, width, and height in feet. This is given by:\[ V = 8.0 \, \text{ft} \times 4.0 \, \text{ft} \times 4.0 \, \text{ft} = 128.0 \, \text{ft}^3 \]

Step 2: Convert Cubic Meters to Cubic Feet

We need to convert 3.0 cubic meters into cubic feet. The conversion factor is 1 cubic meter = 35.3147 cubic feet.\[ 3.0 \, \text{m}^3 \times 35.3147 \, \text{ft}^3/\text{m}^3 = 105.9441 \, \text{ft}^3 \]

Step 3: Calculate the Number of Cords

To find how many cords fit in 3.0 cubic meters, we divide the total cubic feet for 3.0 cubic meters by the cubic feet per cord.\[ \text{Number of cords} = \frac{105.9441 \, \text{ft}^3}{128.0 \, \text{ft}^3/\text{cord}} \approx 0.827 \]

Key Concepts

Cubic Feet

Cubic feet is a measure of volume that is used frequently in the United States. When measuring the volume of something in cubic feet, you are essentially determining how much three-dimensional space it occupies.
This unit is often used in construction, shipping, and storage to define the capacity of a container, room, or even a pile of wood, like a cord.

• A cubic foot is defined as a cube with sides of one foot in length.
• To find the volume represented in cubic feet, you multiply the length, width, and height of the object, all measured in feet.

For example, if you have a box that is 2 feet long, 3 feet wide, and 4 feet high, the volume in cubic feet is calculated by: \[V = 2 \, \text{ft} \times 3 \, \text{ft} \times 4 \, \text{ft} = 24 \, \text{ft}^3\]Converting between units, like cubic feet and cubic meters, is common when dealing with different measurement systems and helps ensure accurate calculations across contexts.

Cubic Meters

Cubic meters is a widely used metric unit for measuring volume and is the standard unit for volume in the International System of Units (SI). It is often utilized in countries that use the metric system rather than imperial measurements like cubic feet.

• A cubic meter is defined as the volume of a cube with side lengths of one meter.

This unit is particularly useful for international shipping, business transactions, and engineering, where precise measurements of volume are crucial. To convert from cubic meters to cubic feet, you use the conversion factor: \[1 \, \text{m}^3 = 35.3147 \, \text{ft}^3\]This factor allows for precise conversion from a cubic meter to the less commonly used cubic foot in certain regions. It's these conversion skills that are essential for solving real-world problems involving volume across different unit systems.

Volume Calculation

Volume calculation expands on the basic principles of measuring how much space a substance, object, or container occupies. It's a fundamental concept in mathematics and science, with practical applications in various fields.

• Volume can be calculated by multiplying the dimensions of an object: length, width, and height.
• For objects with irregular shapes, volume might be calculated via displacement of liquid or using integral calculus for objects with curved surfaces.

Using these basic principles, calculating how many cords of wood fit in a specific volume involves both understanding the measurement units and applying cubic calculations. Because a "cord" of wood is a specific measurement, knowing how to determine its volume in cubic feet is crucial. You can then compare it to another volume, like you would when trying to find how many cords fit into 3 cubic meters. This illustrates how mathematical principles guide practical, everyday tasks and decisions.