Question

Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.

Answer 1

The area of a rectangular rug can be expressed as A = l × w, where 'l' is the length and 'w' is the width. Equivalent expressions can be derived using algebraic properties. Comparing two areas involves calculating the square of their side lengths and forming a ratio.

Step-by-step explanation:

To write an algebraic expression representing the area of a rectangular rug, you need the length and the width of the rectangle. If the length is represented by 'l' and the width by 'w', the area 'A' can be expressed as A = l × w. This is based on the formula for the area of a rectangle.

To write an equivalent expression using properties of operations, you might factor out a common term if 'l' and 'w' are not just simple variables but expressions themselves, or you might expand them if they are factored expressions. For example, if 'l = 3x' and 'w = x + 4', then the area would be A = 3x(x + 4). Applying the distributive property, you would get A = 3x² + 12x, which is an equivalent expression for the area.

Comparing the areas of two squares involves squaring their side lengths and forming a ratio ratio = larger area / smaller area. The area's dimension is squared length (L²), so for the squares with side lengths 's1' and 's2', their areas would be s1² and s2² respectively. The ratio of their areas will then be s1² / s2².

To find the actual area of a rug from a scale drawing, you need to multiply the length and width measured on the drawing by the scale factor. Suppose the drawing's scale factor is 1/24, and the dimensions on the drawing are 2 feet (length) by 1 foot (width). The actual area would be (2 × 24) × (1 × 24) = 48 × 24 square feet.

Answer 2

If
`\( l \)` and
`\( w \)` were given as expressions themselves, such as
`\( l = a + b \)` and
`\( w = c + d \)`, we could use the distributive property to write an equivalent expression, such as
`\( A = (a + b)(c + d) \)`, which would expand to
`\( A = ac + ad + bc + bd \)`.

To write an algebraic expression for the area of a rectangular rug, we need two variables to represent the length and the width of the rug. Let's use
`\( l \)` to represent the length and
`\( w \)` to represent the width.

The formula for the area
`\( A \)` of a rectangle is:

`\[ A = l * w \]`

This is the algebraic expression representing the area of the rectangular rug.

Using properties of operations, we could potentially factor common terms if given more specific terms, or we could expand the expression if it were a product of binomials. However, without additional information on the values of
`\( l \)` and
`\( w \)`, we cannot create an equivalent expression that is significantly different from the original. The expression
`\( A = l * w \)` is the most straightforward representation of the area of a rectangle.

If
`\( l \)` and
`\( w \)` were given as expressions themselves, such as
`\( l = a + b \)` and
`\( w = c + d \)`, we could use the distributive property to write an equivalent expression, such as
`\( A = (a + b)(c + d) \)`, which would expand to
`\( A = ac + ad + bc + bd \)`.