Final answer:
To determine the dimensions of the rectangular garden, an algebraic model can be set up using the equations representing the perimeter and area of the garden. Solving these equations will yield the values for the length and width of the garden. The quadratic equation derived from the area equation can be solved to find the possible values of the width, and then the length can be determined based on these width values.
Step-by-step explanation:
The first step in solving this problem is to set up an algebraic equation to represent the situation. Let's say the length of the garden is L and the width is W. The perimeter of the rectangular garden is 2L + 2W, and we know that it should be equal to 120 feet. So, we can set up the equation 2L + 2W = 120. The area of the garden is given by the formula A = L * W, and we know that it should be equal to 875 square feet. So, we can set up the equation L * W = 875.
To find the dimensions of the rectangular area, we can use these two equations to solve for L and W. We can start by solving the first equation for one variable in terms of the other. For example, we can solve for L in terms of W by subtracting 2W from both sides of the equation: 2L = 120 - 2W. Then, divide both sides of the equation by 2 to solve for L: L = (120 - 2W) / 2.
Next, we substitute this expression for L into the second equation: ((120 - 2W) / 2) * W = 875. Simplify this equation and solve for W by multiplying both sides by 2 and rearranging the terms: W^2 - 60W + 875 = 0. This is a quadratic equation, which can be solved using factoring, completing the square, or the quadratic formula.
Once we find the value(s) of W, we can substitute it back into the equation L = (120 - 2W) / 2 to solve for the corresponding value(s) of L. These values of L and W will represent the dimensions of the rectangular garden.