- Identify the relationship between total bricks, bricks per square foot, and the area of the pavilion as \(A = \frac{N}{B}\).
- Substitute the given values: \(A = \frac{2.967 \times 10^5}{6.9}\).
- Calculate the division: \(\frac{2.967}{6.9} = 0.43\), resulting in \(A = 0.43 \times 10^5\).
- Convert to scientific notation by adjusting the exponent, yielding the final answer \(\boxed{4.3 \times 10^4}\).
Explanation
- Understanding the Relationship and Problem Analysis
The total number of bricks used to build the pavilion is given as \(N = 2.967 \times 10^5\). We are also given that the construction rate is \(6.9\) bricks per square foot. To find the total area of the pavilion in square feet, we denote the total area by \(A\). The relationship between the total number of bricks, the density of bricks (bricks per square foot), and the area is given by the formula:
\[N = B \times A\]
where \(B\) is the number of bricks per square foot. Our goal is to find \(A\), so we rearrange the formula:
\[A = \frac{N}{B}\]
Substituting the given values into this equation:
\[A = \frac{2.967 \times 10^5}{6.9}\]
- Calculating the Total Area and Converting to Scientific Notation
To calculate the total area, we divide the constant terms and keep the power of ten:
\[A = \left( \frac{2.967}{6.9} \right) \times 10^5\]
Dividing \(2.967\) by \(6.9\), we get \(0.43\).
So the expression becomes:
\[A = 0.43 \times 10^5\]
To express this in scientific notation (in the form \(a \times 10^n\) where \(1 \leq a < 10\)), we rewrite \(0.43\) as \(4.3 \times 10^{-1}\):
\[A = 4.3 \times 10^{-1} \times 10^5\]
Using the laws of exponents, we add the exponents of 10:
\[A = 4.3 \times 10^{-1 + 5} = 4.3 \times 10^4\]
The total square footage of the pavilion is \(4.3 \times 10^4\).
Examples
This type of calculation is common in construction and architecture. For example, if you know the total number of materials used and how many items are required per unit of measurement, you can determine the total extent or dimensions of a project. Whether it is calculating the total floor space of a building based on tiling, or estimating the total area covered by a certain number of bricks, these algebraic relations allow for precise planning and material management in real-world infrastructure projects.