To solve this problem, we can use the given information to set up a system of equations. Here's the breakdown:

Step 1: Define Variables

• Let $L$ be the length of the rectangular field (the side parallel to the river).
• Let $W$ be the width of the rectangular field (the two sides perpendicular to the river).

Step 2: Set up Equations

• The total length of fencing used is 53 meters for three sides (two widths and one length), so: $2W + L = 53$
• The area of the land is 306 square meters, so: $L \times W = 306$

Step 3: Solve for One Variable

From the first equation, solve for $L$: $L = 53 - 2W$

Step 4: Substitute into the Area Equation

Substitute $L = 53 - 2W$ into the area equation: $(53 - 2W) \times W = 306$

Step 5: Expand and Solve the Quadratic Equation

$53W - 2W^2 = 306$ $2W^2 - 53W + 306 = 0$

Now, use the quadratic formula to solve for $W$: $W = \frac{-(-53) \pm \sqrt{(-53)^2 - 4 \times 2 \times 306}}{2 \times 2}$ $W = \frac{53 \pm \sqrt{2809 - 2448}}{4}$ $W = \frac{53 \pm \sqrt{361}}{4}$ $W = \frac{53 \pm 19}{4}$

Step 6: Calculate the Possible Values for $W$

$W = \frac{53 + 19}{4} = \frac{72}{4} = 18$ $W = \frac{53 - 19}{4} = \frac{34}{4} = 8.5$

Step 7: Find the Corresponding $L$ Values

Now, substitute the possible values of $W$ back into $L = 53 - 2W$:

• For $W = 18$: $L = 53 - 2(18) = 53 - 36 = 17$
• For $W = 8.5$: $L = 53 - 2(8.5) = 53 - 17 = 36$

Final Answer:

The two possible sets of dimensions for the field are:

• $L = 17$ meters, $W = 18$ meters
• $L = 36$ meters, $W = 8.5$ meters

Would you like more details on the calculations or any clarification?

Here are 5 related questions:

1. How would the dimensions change if the fencing length were 60 meters?
2. What if the area was larger, say 400 square meters?
3. How can we express the area in terms of $W$ and the fencing constraints?
4. What happens if the shape of the field were different, like a triangle?
5. Can this type of problem be solved graphically using a parabola?

Tip: Always verify quadratic roots by substituting them back into the original equation to ensure accuracy.