Question

juan wants to change the shape of his vegetable garden from a square to a rectangle, but keep the same area so he can grow the same amount of vegetables. the rectangular garden will have a length that is 2 times the length of the square garden, and the width of the new garden will be 16 feet shorter than the old garden. the square garden is x feet by x feet. what is the quadratic equation that would model this scenario?
\\( x^2 = (2x)(x - 16) \\)
\\( x^2 = (x)(x - 16) \\)
\\( x^2 = (x)(x + 16) \\)
\\( x^2 = (2x)(x + 16) \\)

Explanation:

Step1: Find area of square garden

The square garden has side length \( x \) feet, so its area is \( x \times x = x^2 \) square feet.

Step2: Find length and width of rectangular garden

The length of the rectangular garden is 2 times the side length of the square, so length \( = 2x \) feet. The width of the rectangular garden is 16 feet shorter than the side length of the square, so width \( = x - 16 \) feet.

Step3: Find area of rectangular garden

The area of a rectangle is length times width, so area of rectangular garden is \( (2x)(x - 16) \) square feet.

Step4: Set areas equal (since area is same)

Since the area of the square and rectangle are equal, we set \( x^2=(2x)(x - 16) \).

Answer:

\( x^2 = (2x)(x - 16) \) (the first option)