Question

the garden in ms. barton’s backyard is shaped like a rectangle that is 9 feet longer than it is wide. the width of the garden is x feet. which of the following quantities may be represented by the expression ((x + 2)(x + 10))?

a. the perimeter, in feet, of ms. barton’s garden if she made it 2 feet wider and 1 foot longer

b. the area, in square feet, of ms. barton’s garden if she made it 2 feet wider and 10 feet longer

c. the area, in square feet, of ms. barton’s garden if she made it 2 feet wider and 1 foot longer

d. the perimeter, in feet, of ms. barton’s garden if she made it 2 feet wider and 10 feet longer

Explanation:

Step1: Analyze original garden dimensions

The width of the original garden is \( x \) feet. The length is 9 feet longer than the width, so the original length is \( x + 9 \) feet.

Step2: Analyze modified dimensions for each option

• Option A and D (Perimeter): The formula for the perimeter of a rectangle is \( P = 2(\text{length} + \text{width}) \), which is a linear expression (degree 1), but \( (x + 2)(x + 10) \) is a quadratic expression (degree 2), so A and D (perimeter options) can be eliminated.
• Option B: If we make the garden 2 feet wider, the new width is \( x + 2 \). If we make it 10 feet longer, the new length would be \( (x + 9) + 10 = x + 19 \), not \( x + 10 \). So B is incorrect.
• Option C: If we make the garden 2 feet wider, the new width is \( x + 2 \). If we make it 1 foot longer, the new length is \( (x + 9) + 1 = x + 10 \). The area of a rectangle is \( \text{length} \times \text{width} \), so the new area is \( (x + 2)(x + 10) \), which matches the given expression.

Answer:

C. the area, in square feet, of Ms. Barton’s garden if she made it 2 feet wider and 1 foot longer