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?

Answer 1

The quadratic equation that would model the scenario where Juan changes his square vegetable garden into a rectangular garden with the same area is 0 = -x^2 + 32x.

Step-by-step explanation:

Juan is changing his square vegetable garden into a rectangular garden and wants to retain the same area. If the square garden is x feet by x feet, its area is x squared (x^2). The new rectangular garden will have a length that is double the side of the square (2 feet) and a width that is 16 feet shorter (x - 16 feet). Thus, the area of the rectangle is (2x) * (x - 16). To keep the same area, the area of the square must be equal to the area of the rectangle, resulting in the equation:

x^2 = 2x(x - 16)

Expanding the right side gives us the equation 2x^2 - 32x, which can be set to zero to find x by using the quadratic formula:

x^2 - 2x^2 + 32x = 0

The quadratic equation modeling this scenario is:

0 = x^2 - 2x^2 + 32x

Or simplified:

0 = -x^2 + 32x

Answer 2

For the square, the area will be 4x4, which will equal 16 sq ft. Since Juan is wanting to turn it to a rectangle, he will have to use the equation 8x2.