Question

The perimeter of a rectangle is 44 inches. The area of the rectangle is

120 square inches. If the width of the rectangle is x inches, determine
the length of the rectangle using an algebraic solution.

Answer 1

The length of the rectangle is 10 inches.

Let's denote the length of the rectangle as
`\( L \)` and the width as
`\( x \)`. The perimeter of a rectangle is given by the formula
`\( P = 2L + 2W \)`, and the area is given by the formula
`\( A = LW \)`.

Given that the perimeter is 44 inches, we have:

`\[ P = 2L + 2x = 44 \]`

And given that the area is 120 square inches, we have:

`\[ A = LW = 120 \]`

Now, let's use these equations to solve for
`\( L \)`.

1. Express
`\( L \)` in terms of
`\( x \)` from the perimeter equation:

`\[ 2L + 2x = 44 \]`

`\[ 2L = 44 - 2x \]`

`\[ L = (44 - 2x)/(2) \]`

`\[ L = 22 - x \]`

2. Substitute the expression for
`\( L \)` into the area equation:

`\[ LW = 120 \]`

`\[ (22 - x)x = 120 \]`

3. Solve the quadratic equation:

`\[ 22x - x^2 = 120 \]`

`\[ x^2 - 22x + 120 = 0 \]`

4. Factor the quadratic equation:

`\[ (x - 10)(x - 12) = 0 \]`

This equation has two solutions:
`\( x = 10 \)` and
`\( x = 12 \)`. However, since the width cannot be negative, we discard the
`\( x = 10 \)` solution.

5. Substitute the remaining value of
`\( x \)` back into the expression for
`\( L \)`:

`\[ L = 22 - x \]`

`\[ L = 22 - 12 \]`

`\[ L = 10 \]`

So, the length of the rectangle is 10 inches.