To determine the length and width of a rectangle given its area and the relationship between its length and width, you can use algebraic equations.
Let's say:
- \( A \) is the area of the rectangle.
- \( l \) is the length.
- \( w \) is the width.
You know that the area of the rectangle is given by the formula: \[ A = l \times w \]
If a relationship between the length and width is given, such as "the length is twice the width," you can express one variable in terms of the other. For example, if \( l = 2w \), you would substitute this into the area formula.
So, the area equation becomes: \[ A = (2w) \times w = 2w^2 \]
To find \( w \), you would rearrange this equation: \[ w^2 = \frac{A}{2} \] \[ w = \sqrt{\frac{A}{2}} \]
Once you find \( w \), you can calculate \( l \): \[ l = 2w \]
In general, if you have a different relationship, such as \( l = kw \) (where \( k \) is a constant), you would substitute that into the area formula: \[ A = (kw) \times w = kw^2 \] Then solve for \( w \): \[ w^2 = \frac{A}{k} \] \[ w = \sqrt{\frac{A}{k}} \] And subsequently find \( l \): \[ l = kw \]
Using this method allows you to find the dimensions of the rectangle based on its area and the specified relationship between length and width.