Question

What is the mathematical formula used to solve the problem of determining the length and width of a rectangle, given its area and the relationship between its length and width?.

Answer 1

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.