Question

Katie’s old bedroom was shaped like a rectangle. It had a length that was 4 times its width. When katie’s family moved her new bedroom was also shaped like a rectangle it was 5 feet longer and 4 feet wider than her old bedroom if w represents the width of katies old bedroom which expression represents the difference between the same area of her new bedroom and the area of her old bedroom

Answer 1

The difference in area between Katie's new bedroom and her old bedroom is expressed as the difference between the area of the new bedroom and that of the old bedroom. The expression for this difference is 25w + 20, where w is the width of the old bedroom.

Step-by-step explanation:

The question asks us to determine the difference in area between Katie's old bedroom and her new bedroom, using the variable w to represent the width of Katie's old bedroom. Since the old bedroom is rectangular with a length that is four times its width, then the area of the old bedroom is w × (4w) = 4w^2. Katie's new bedroom is 5 feet longer and 4 feet wider than her old bedroom, so its dimensions are (w + 4) for the width and (4w + 5) for the length. Therefore, the area of the new bedroom is (w + 4)(4w + 5). The difference in area between the new and old bedrooms is (w + 4)(4w + 5) - 4w^2.

To find this expression, we need to expand the binomial for the new bedroom's area and subtract the old bedroom's area from it. This results in:

new area = (w + 4)(4w + 5) = 4w^2 + 20w + 5w + 20 = 4w^2 + 25w + 20

old area = 4w^2

Difference = (new area) - (old area) = (4w^2 + 25w + 20) - 4w^2 = 25w + 20

This gives us the expression representing the area difference between Katie's new bedroom and her old bedroom.