This is an optimization problem where we aim to minimize the fencing required for a rectangular property of area $5190$ square yards, with one side adjacent to a cliff. Since only three sides need fencing, we’ll set up the problem using calculus to find the minimum fencing length.

Let:

• $x$ be the width (parallel to the cliff, no fencing needed on this side).
• $y$ be the length (the two sides perpendicular to the cliff).

Step 1: Set up the area equation

Since the area of the rectangle is $5190$ square yards: $x \cdot y = 5190$ This can be rearranged to express $y$ in terms of $x$: $y = \frac{5190}{x}$

Step 2: Set up the fencing equation

Since only three sides need fencing, the fencing length $F$ will include two sides of length $y$ and one side of length $x$: $F = 2y + x$ Substituting $y = \frac{5190}{x}$: $F = 2\left(\frac{5190}{x}\right) + x$ $F = \frac{10380}{x} + x$

Step 3: Minimize the fencing function

To find the minimum value of $F$, we take the derivative of $F$ with respect to $x$ and set it to zero: $\frac{dF}{dx} = -\frac{10380}{x^2} + 1$ Setting $\frac{dF}{dx} = 0$: $-\frac{10380}{x^2} + 1 = 0$ $\frac{10380}{x^2} = 1$ $x^2 = 10380$ $x = \sqrt{10380} \approx 102$

Step 4: Solve for $y$

Substitute $x = 102$ back into the area equation: $y = \frac{5190}{102} \approx 51$

Step 5: Calculate the total fencing

The minimum fencing required is: $F = 2y + x = 2(51) + 102 = 204 \text{ yards}$

Final Answers:

• Least amount of fencing required: 204 yards.
• Two sides should be: 51 yards each.
• One side should be: 102 yards.

Would you like further details or have any questions?

1. What if the area of the property changed? How would the calculations adjust?
2. How would adding a fence on the fourth side alter the required fencing length?
3. Why does minimizing one variable affect the total length of fencing?
4. Could this solution be applied to a non-rectangular property with similar conditions?
5. How would changing the cliff side affect the fencing needed?

Tip: In optimization problems, always express one variable in terms of the other using given constraints before differentiating to minimize or maximize.