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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.

A rancher is planning to build a rectangular fence around his property, with one side of the property being next to a cliff. The property is 5190 square yards, and only the three sides not adjacent to the cliff will be fenced off. What is the least amount of fencing required for this job? How long and wide should the park be for the fencing to be minimized?

Discover how to minimize fencing for a 5190-square-yard ranch property with one side along a cliff. This problem involves optimization using calculus to determine the minimum fencing needed and the ideal dimensions for the fenced area.

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