Problem 100 - Algebra and Trigonometry

Question

Mowing a Field A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals (see the figure). The field measures b feet by b feet, and the mowed strip is x feet wide. (a) Explain why the area of the mowed portion is \(b^{2}-(b-2 x)^{2}\) (b) Factor the expression in part (a) to show that the area of the mowed portion is also \(4 x(b-x) .\)

Step-by-Step Solution

Answer

The area of the mowed portion is \(b^2 - (b-2x)^2 = 4x(b-x)\).

Step 1: Understanding the Problem

We are given a square field with side length \(b\) feet and a mowed strip of width \(x\) feet around the edges. Our task is to understand the problem correctly in order to calculate the area of the mowed portion.

Step 2: Identify the Dimensions

The area of the entire field is \(b^2\). However, the unmowed section forms a smaller square in the center with side length \(b-2x\) because \(x\) feet are mowed from each side of the field's border.

Step 3: Calculate the Area of the Mowed Portion (Part a)

The area of the mowed portion is the area of the entire field minus the area of the unmowed portion. This is given by:\[A_{\text{mowed}} = b^2 - (b-2x)^2\]

Step 4: Simplify the Expression (Part a)

To simplify \(b^2 - (b-2x)^2\), use the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\). Substitute \(a = b\) and \(b = b-2x\):\[A_{\text{mowed}} = [b - (b - 2x)][b + (b - 2x)] = 2x(2b - 2x) = 4x(b-x)\]

Step 5: Factor to Confirm the Expression (Part b)

We factored the expression as required to confirm that the area of the mowed portion is indeed \(4x(b-x)\), demonstrating that both expressions are equivalent.

Key Concepts

Difference of Squares

The concept of the difference of squares is a handy tool in algebra, and plays a crucial role in our problem of the square field. It refers to an expression of the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\). Let's break it down further.

• Consider two squares where one has side length \(a\) and the other has side length \(b\).
• Their areas will be \(a^2\) and \(b^2\), respectively.
• The difference between these two areas is \(a^2 - b^2\), which geometrically represents the area outside the smaller square but within the larger square.

In our problem, we used this formula to express the mowed area: \(b^2 - (b-2x)^2\). Substituting \(b\) as \(a\) and \(b-2x\) as \(b\) into the formula, gives us the factorization: \((b - (b-2x))(b + (b-2x))\). This simplifies further into \(4x(b-x)\), as verified in the solution.

Factoring

Factoring is a key algebraic technique used to simplify expressions and solve equations by expressing a polynomial as a product of its factors.

• Finding the factors of an expression like \(b^2 - (b-2x)^2\) is essential to reduce complexity and solve related problems easily.
• In our exercise, we demonstrated this by simplifying the difference of squares into \((b - (b-2x))(b + (b-2x))\), which further simplified to \(4x(b-x)\).
• This shows how the area of the mowed part can be expressed and understood simply as a product of its dimensions.

Factoring not only helps us to understand the problem better but also aids in confirming the given expressions. Recognizing patterns like the difference of squares can greatly simplify algebraic manipulation.

Geometry Problems

Our problem of mowing a square field involves both algebra and geometry principles, showcasing their deep connection. Geometry problems often use algebra to solve for areas and dimensions effectively.

• The entire field is a perfect square, an important geometric shape with properties that simplify calculations. The unmowed part is another square nested inside the larger one.
• This gives rise to challenges of calculating specific areas, which we solve using algebraic expressions.
• Understanding the layout—how the mowed strip surrounds the internal square—is crucial for setting up correct equations.

By relating geometric figures to algebraic expressions, we can effortlessly compute the area of complex shapes, just as done with the field’s mowed region. This fusion of fields demonstrates the practicality of mathematical concepts across different scenarios.