91Ó°ÊÓ

The area of a rectangle is a basic concept in geometry, crucial for solving many problems like the one in this exercise. A rectangle has four sides, with opposite sides being equal in length. Its area can be determined using the formula:

• \( ext{Area} = ext{length} imes ext{width} \)

This simple multiplication gives the amount of space within the rectangle's boundaries. In practical terms, imagine the area as the number of square units that cover a flat surface.

In this specific problem, the area is given as \( 128 \, ext{ft}^2 \). You need to find the dimensions of the rectangle that fit these criteria, considering additional conditions provided. When faced with any area calculation problem, always start by identifying the length and width parameters clearly. This will form the basis of setting up your equation for solving.

A rectangular enclosure typically involves surrounding a specific area using boundaries, which can be fences, walls, or other materials. These are common real-world problems that require understanding both geometry and optimization, like in this problem where one side of the rectangle is part of a barn.

In fencing or enclosing a rectangle where one side is constrained (here, the barn), you must optimize the use of the remaining sides. The key aspect is to effectively use the available resources to maximize or specify the desired area, as given in the problem statement.

• One side of the rectangle is twice as long as an adjacent side.
• This creates a practical optimization problem where one must decide how to configure the remaining three sides ensuring a specific area.

Understanding the relationship between side lengths is crucial in setting up the problem. Efficient use of material (fencing) involves calculating only the necessary lengths needed while adhering to the area requirements.

Quadratic equations appear frequently in optimization problems, especially in dealing with areas like rectangles or curves. In the given exercise, solving a quadratic equation is necessary to find the length of the rectangle's sides.

The problem simplifies to the equation:

• \( 2x^2 = 128 \)

This equation is derived from the area formula. Dividing through gives:

• \( x^2 = 64 \)

Next, take the square root of both sides to isolate \( x \):

• \( x = 8 \)

The solution tells us that one side, perpendicular to the barn, is 8 feet. Solving quadratic equations often involves:

• Rearranging the equation to a standard form \( ax^2 + bx + c = 0 \)
• Solving for \( x \) using factorization, completing the square, or the quadratic formula.

These are essential techniques in algebra when working on enclosed rectangular problems like this one.

Fencing problems blend geometry with optimization and practical application. They generally involve finding the best configuration of fencing material to enclose a particular space, often at minimal cost or with material constraints.

In the example problem:

• Part of the barn acts as one side of the rectangle, so only three sides require fencing.
• The calculated fencing is needed for sides measuring 8 feet, 8 feet, and 16 feet, as derived from the dimensions of the rectangle.

The total fencing requirement becomes:

• \( 8 + 8 + 16 = 32 \, ext{feet} \)

When tackling fencing problems, always consider the relationship between the dimensions of the area and available materials. This ensures that no resources are wasted and the enclosure meets the criteria for the specific application. Thus, you can achieve the desired geometric structure using practical and mathematical reasoning.