91Ó°ÊÓ

Understanding the dimensions of a rectangle is a fundamental concept in geometry. A rectangle has two pairs of opposite sides that are equal in length. When referring to the dimensions of a rectangle, we talk about its width (shorter side) and length (longer side). Here, the length is defined as 6 units more than the width, which means if we know either dimension, we can easily find the other.
For example, if we denote the width by the variable \(w\), the length \(l\) will be expressed as \(l = w + 6\). This helps simplify solving related mathematical problems as it provides a direct relationship between the two dimensions.

The perimeter of a rectangle is the total distance around the outer edge of the rectangle. It is calculated by adding together the lengths of all four sides. For a rectangle with length \(l\) and width \(w\), the perimeter \(P\) is given by the formula: \[P = 2l + 2w \].
In our problem, substituting \(l = w + 6\) into the perimeter formula provides us with the original perimeter of the rectangle: \[P_{original} = 2(w + 6) + 2w = 4w + 12 \]. This equation will be helpful later when comparing to the new perimeter after changing dimensions.
To find the new dimensions, we need to account for the changes: the width increases by 10 units (\(w + 10\)) and the length is tripled (\(3(w + 6)\)). Substituting these new dimensions back into the formula provides the new perimeter: \[P_{new} = 2 \times 3(w + 6) + 2 \times (w + 10) = 8w + 56 \].This step is crucial for understanding how the perimeter changes with the dimensions.

Algebraic equations are used to find unknown values by translating word problems into mathematical expressions. In this exercise, we used algebra to formulate and solve equations to find the dimensions of the rectangle based on the given conditions.
First, we defined the relationship between length and width as \(l = w + 6\) and applied the perimeter formula to set up expressions for the original and new perimeters: \[P_{original} = 4w + 12 \] and \[P_{new} = 8w + 56 \].
Next, we used the given condition that the new perimeter is 56 units more than the original perimeter. This gives us: \[8w + 56 = 4w + 12 + 56 \]. Simplifying this, we get \[8w = 4w + 68 \] which further simplifies to \[4w = 12 \], leading to the solution \(w = 3\).
Substituting \(w = 3\) back into the length equation, we find \(l = 9\). Therefore, the original dimensions of the rectangle are a width of 3 units and a length of 9 units.Understanding how to set up and solve such algebraic equations is essential for tackling a wide range of mathematical problems involving rectangles and other geometric shapes.