Finding Driveway Width An Algebraic Solution
In the realm of practical mathematics, everyday scenarios often present opportunities to apply algebraic concepts. Consider this problem involving Sharon's driveway, where we'll delve into the fascinating interplay of area, length, and width. Our task is to determine the width of the driveway, given its area and length, and in this comprehensive exploration, we will not only solve the problem but also unpack the underlying mathematical principles, providing valuable insights for anyone seeking to sharpen their algebraic problem-solving skills.
Deciphering the Driveway Dimensions: A Step-by-Step Approach
To find the width of Sharon's driveway, we'll leverage the fundamental relationship between the area, length, and width of a rectangle: Area = Length × Width. In this scenario, we're provided with the area and length, expressed as algebraic expressions, and our mission is to determine the width, also as an algebraic expression. This involves employing the concepts of polynomial factorization and algebraic division, showcasing the real-world applications of these mathematical tools.
1. Expressing the Given Information
Let's begin by meticulously outlining the information we have at our disposal:
- Area of the driveway: 5x² + 43x - 18
- Length of the driveway: x + 9
- Our objective: Find the width of the driveway
2. Applying the Area Formula
As mentioned earlier, the area of a rectangle is the product of its length and width. We can represent this mathematically as:
Area = Length × Width
In our case, we can substitute the given values:
5x² + 43x - 18 = (x + 9) × Width
3. Isolating the Width
To isolate the width, we need to perform an algebraic operation that undoes the multiplication by the length (x + 9). This is achieved through division. We divide both sides of the equation by (x + 9):
Width = (5x² + 43x - 18) / (x + 9)
Now, our task is to simplify the expression on the right-hand side, which involves dividing the quadratic expression (5x² + 43x - 18) by the linear expression (x + 9).
4. Polynomial Long Division or Factoring
There are two primary methods to tackle this division: polynomial long division or factoring. Let's explore both approaches.
Method 1: Polynomial Long Division
Polynomial long division is a systematic technique for dividing polynomials. It mirrors the familiar process of long division with numbers. Here's how it unfolds in this context:
5x - 2
x + 9 | 5x² + 43x - 18
-(5x² + 45x)
-----------------
-2x - 18
-(-2x - 18)
-----------
0
The quotient obtained from the long division is 5x - 2, which represents the width of the driveway.
Method 2: Factoring
Factoring involves expressing a polynomial as a product of simpler polynomials. In this case, we aim to factor the quadratic expression 5x² + 43x - 18. To do this, we seek two binomials whose product equals the quadratic. Let's break down the process:
-
Identify the factors of the leading coefficient (5) and the constant term (-18).
- Factors of 5: 1 and 5
- Factors of -18: ±1, ±2, ±3, ±6, ±9, ±18
-
Experiment with different combinations of factors to find a pair that, when multiplied using the FOIL method (First, Outer, Inner, Last), yields the original quadratic expression.
After some trial and error, we discover the following factorization:
5x² + 43x - 18 = (5x - 2)(x + 9)
Now, we can rewrite the expression for the width:
Width = (5x² + 43x - 18) / (x + 9) = [(5x - 2)(x + 9)] / (x + 9)
We can cancel out the common factor (x + 9) from the numerator and denominator:
Width = 5x - 2
5. The Solution
Both polynomial long division and factoring lead us to the same result: the width of Sharon's driveway is 5x - 2.
Solidifying Understanding: Key Concepts and Implications
This problem serves as a powerful illustration of how algebraic concepts can be applied to solve real-world scenarios. Let's distill the key takeaways and insights:
- Area of a rectangle: The foundation of the problem lies in the formula Area = Length × Width. Understanding this relationship is crucial for solving geometric problems involving rectangles.
- Polynomial factorization: Factoring quadratic expressions is a fundamental algebraic skill with wide-ranging applications. It allows us to simplify expressions, solve equations, and gain insights into the structure of polynomials.
- Polynomial long division: Polynomial long division provides a systematic method for dividing polynomials, even when factoring is not straightforward. It's a versatile tool in algebraic manipulation.
- Connecting algebra and geometry: This problem beautifully bridges the gap between algebra and geometry, showcasing how algebraic expressions can represent geometric quantities and how algebraic operations can be used to solve geometric problems.
Expanding Horizons: Additional Practice and Exploration
To further solidify your understanding and hone your problem-solving skills, consider tackling similar problems. Here are some avenues for exploration:
- Variations in the given information: Explore scenarios where you're given the area and width and need to find the length, or where you're given the perimeter and one dimension and need to find the other.
- More complex polynomial expressions: Challenge yourself with problems involving higher-degree polynomials or more intricate factorizations.
- Real-world applications: Seek out real-world problems that involve area, length, width, and polynomial expressions. This will deepen your appreciation for the practical relevance of algebraic concepts.
Conclusion: The Power of Mathematical Reasoning
Solving Sharon's driveway problem has been more than just finding a numerical answer; it's been an exercise in mathematical reasoning, problem-solving strategies, and the interconnectedness of algebraic and geometric concepts. By mastering these skills, we empower ourselves to tackle a wide range of real-world challenges with confidence and precision. The width of Sharon's driveway, 5x - 2, stands as a testament to the power of mathematical thinking.
