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Chapter 4: Problem 46

Explain why a square foot contains 144 square inches.

Short Answer

Expert verified
A square foot contains 144 square inches because a square foot is an area with the dimensions of 1 foot by 1 foot, and there are 12 inches in a foot. When dividing a square foot into square inches, it forms a grid with 12 rows and 12 columns. The total number of squares (square inches) in this grid can be calculated by multiplying the number of rows by the number of columns, which is 12 脳 12, resulting in 144 square inches. Hence, \( 1 \, \text{sq ft} = 144 \, \text{sq in} \).

Step by step solution

01

Convert linear units

First, we need to know the relationship between inches and feet in terms of linear units. We can establish that there is 1 foot (ft) equal to 12 inches (in), as a standard measurement.
02

Understand area units

Now, let's think about area units. A square foot (sq ft) is an area that has the dimensions of one foot by one foot. A square inch (sq in) is an area that has the dimensions of one inch by one inch.
03

Calculate the number of square inches in a square foot

Since there are 12 inches in a foot, there will be 12 rows (each of the length of 1 foot) when a square foot is divided into square inches horizontally. Similarly, there will be 12 columns (each of the length of 1 foot) when dividing vertically. So, when you divide a square foot into square inches, you will have a grid with 12 rows and 12 columns. To find the total number of squares in this grid, you can use multiplication: Total squares (square inches) in a square foot = number of rows 脳 number of columns = 12 脳 12.
04

Conclusion

So, there are 144 square inches in a square foot: \( 1 \, \text{sq ft} = 12 \times 12 = 144 \, \text{sq in} \) This shows that a square foot contains 144 square inches, as required in the exercise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding a Square Foot
A square foot is a unit of area that is commonly used in the United States. It's particularly useful in real estate, architecture, and many other fields.
A square foot represents the area within a square with each side measuring exactly one foot in length. This concept of measuring area is very straightforward, as it involves the space enclosed within the boundary of the square.
Key points to remember about a square foot:
  • It is a unit representing area, not length.
  • Its sides are all equal in length, each being one foot.
  • The total area of a square foot can be calculated as length 脳 width, which equals 1 foot 脳 1 foot.
Understanding square feet is crucial for dealing with larger areas, allowing one to know how much space is available, for example, in a room or a piece of land.
The Concept of a Square Inch
A square inch is, much like a square foot, a unit of area measurement. However, it covers a much smaller space given the transition from feet to inches.
A square inch is the area within a square where each side measures exactly one inch long. This smaller unit of area is often used in more detailed or smaller scale tasks.
Important aspects of a square inch:
  • This measurement is primarily used for smaller surfaces or when precision is more critical.
  • It simplifies comparisons between surfaces, such as pieces of paper or small screens, where square feet would be too large a measure.
  • Understanding square inches begins similarly, by recognizing each side's length as one inch.
Square inches help bring precision in measurement, especially when fine details or smaller dimensions matter.
Grasping Unit Conversion
Unit conversion is a vital math skill. It is about changing a quantity expressed in one set of units into another set of units, while maintaining the same quantity.
For area, converting units involves understanding the linear conversion and then extending it to a two-dimensional space. In our case, to convert between square feet and square inches, you need first to know how to convert feet to inches (1 foot = 12 inches).
Key aspects of unit conversion:
  • Learn the relationship between various units (such as feet and inches).
  • Apply the conversion factor properly - if converting one-dimensional units, apply the factor once. For two-dimensional (area) units, apply it twice.
  • Maintain the logical consistency of your calculations to ensure accurate results.
Mastering unit conversion ensures you can accurately calculate and compare areas across different measurement systems.
Multiplication of Units for Area Computation
Multiplying units is a straightforward yet fundamental process in converting and computing areas. It involves using multiplication to expand or convert smaller units to larger unit equivalents.
In the problem given, the multiplication process involves using the number of square inches contained in one row and one column to find the total number of square inches in a square foot.
Basic principles of multiplication of units:
  • You determine how many times a unit fits into another by multiplication.
  • The problem expands from a linear conversion (feet to inches) to cover a two-dimensional area (square feet to square inches).
  • In area calculations, if each side of an area is divided into 12 smaller parts (for inches in a foot), then the total number of parts is 12 脳 12.
This approach solidifies your understanding of how multiplication plays a pivotal role in accurately determining and converting areas.

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