Problem 48 - Algebra 1: Concepts and Skills

Question

The length of a rectangular plot of land is 24 meters more than its width. A paved area measuring 8 meters by 12 meters is placed on the plot. The area of the unpaved part of the land is then 880 square meters. If w represents the width of the plot of land in meters, which of the following equations can be factored to find the possible values of w? HINT: Begin by drawing and labeling a diagram. $$ a.\quad w^{2}+24 w=880 $$ $$ b.\quad w^{2}+24 w+96=880 $$ $$ c.\quad w^{2}+24 w-96=880 $$ $$ d.\quad w^{2}+24 w=96 $$

Step-by-Step Solution

Answer

The equation representing the situation described in the question is \(w^{2} + 24w - 976 = 0\). Here, none of the options given in a, b, c, and d directly match the derived equation. But if 976 in the derived equation is factored as \(32*30 = 880 + 96\), option a) \(w^{2} + 24w = 880\) can be loosely interpreted as similar if you separately account for the 96 square meters of the paved area.

Step 1: Understanding the problem

The question describes a rectangular plot whose length is 24 meters more than the width. A smaller rectangular paved area measuring 8 meters by 12 meters is placed on the plot, leaving an unpaved area of 880 square meters. We have to establish an equation that represents the given scenario where 'w' represents the width of the whole plot.

Step 2: Formulating the equation

The total area of the rectangular plot is the sum of the paved area and the unpaved area. The width of the land is w, and the length is w+24 meters. The area of any rectangle can be found by the product of its length and width. So, the area of the whole plot is w*(w+24) square meters. The area of the paved rectangle is 8*12 = 96 square meters. The area of the unpaved part of the land is 880 square meters. So the full area of the land is equal to 96 + 880 = 976 square meters. Therefore, the following equation arises from these: w*(w+24) = 976.

Step 3: Solving the equation

Expanding this to a quadratic form gives \(w^{2} + 24w = 976\). To make comparison easier with the provided options, let's rearrange this to have zero on one side, so we get \(w^{2} + 24w - 976 = 0\). Now, this equation can be used to find the possible values of w.

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra that help us find unknown values in various problems, especially when dealing with area and other quadratic scenarios. A quadratic equation is typically in the form \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. In many real-world problems, like the one given in the exercise, quadratic equations arise naturally from geometric relationships. In our problem, we derived the equation \[ w^2 + 24w - 976 = 0 \] by equating the product of the length and the width of the rectangular plot to the total area minus the area of the paved part. Solving quadratic equations can be done by various methods such as **factoring**, **completing the square**, or using the **quadratic formula**. Each method serves to find the potential values for \( x \) that satisfy the equation.

Rectangular Area Problems

Rectangular area problems are common in mathematics and involve calculating the space within a rectangular boundary. The basic principle is to multiply the length by the width to get the total area. In real-life situations, like landscaping or architecture, it is essential to understand these concepts to plan spaces efficiently.

In the presented problem, we faced a situation involving a larger rectangular plot with a smaller rectangle carved out as a paved section. We computed the area of the entire plot as \[ w \times (w + 24) \] where \( w \) is the width and \( w + 24 \) is the length. The paved area, measuring **8 meters by 12 meters**, contributed 96 square meters to the total, subtracting from the available unpaved region of 880 square meters. By understanding these areas separately, we were able to construct a quadratic equation that models the situation. Such exercises demonstrate the practical application of rectangle area calculations in solving complex spatial problems.

Factoring Equations

Factoring equations is a technique used to simplify quadratic equations or solve them by breaking them down into simpler components. It involves writing the equation as a product of its factors, thereby revealing the roots or solutions. The standard form of factoring is identifying two binomials whose product gives the original quadratic equation.

In our problem, once arranged as \[ w^2 + 24w - 976 = 0 \], the goal is to factor this expression if possible. The factoring process might require identifying numbers that multiply to give \(-976\) and add to \(24\). This method simplifies solving equations by allowing us to set each factor equal to zero and solve for \( w \). While not all quadratics can be readily factored, many can, and factoring presents a straightforward path to finding potential solutions.