Final answer:
The rectangular patch with a perimeter of 44 meters and an area of 120 square meters has dimensions that can either be 10 meters by 12 meters or 12 meters by 10 meters, as both configurations satisfy the given conditions.
Step-by-step explanation:
A student has a rectangular patch with a perimeter of 44 meters and an area of 120 square meters. To find the dimensions of the rectangle, we'll use the formulas for perimeter and area of a rectangle, which are P = 2l + 2w (where P is the perimeter, l is the length, and w is the width), and A = lw (where A is the area).
Let's denote the length as l and the width as w. The given perimeter, P = 44, gives us the equation 2l + 2w = 44. Similarly, the given area, A = 120, results in lw = 120. We can now solve this system of equations to find the values of l and w.
First, simplify the perimeter equation: l + w = 22. We can then express w as w = 22 - l. Substituting this into the area equation lw = 120 gives l(22 - l) = 120. Expanding and simplifying gives us a quadratic equation: l2 - 22l + 120 = 0. Factoring this quadratic equation, we get (l - 10)(l - 12) = 0, indicating that l = 10 or l = 12.
If l = 10, then w = 22 - 10 = 12, and if l = 12, then w = 22 - 12 = 10. Therefore, the dimensions of the rectangle can either be 10 meters by 12 meters or 12 meters by 10 meters.