A farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). The pasture must contain 125,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river? x = m m y =
Unitary Method
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### Minimizing Fencing for a Rectangular Pasture Adjacent to a River
A farmer plans to enclose a rectangular pasture adjacent to a river. (See the figure below.) The pasture must contain 125,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?
Calculate the dimensions (x and y) where:
\[ x = \_\_\_\_\_ \, \text{m} \]
\[ y = \_\_\_\_\_ \, \text{m} \]
#### Description of the Diagram:
- The diagram shows a rectangular pasture with one side adjacent to a river.
- The sides of the pasture perpendicular to the river are labeled 'y' meters each.
- The side of the pasture parallel to the river, which does not require fencing, is labeled 'x' meters.
- The area of the pasture is \( x \times y = 125,000 \) square meters.
### Solution Process:
1. **Set Up the Equations:**
- The area equation is:
\[ x \times y = 125,000 \]
- The perimeter (amount of fencing) equation is:
\[ P = 2y + x \]
2. **Minimize Fencing Required:**
- Express \( y \) in terms of \( x \) using the area equation:
\[ y = \frac{125,000}{x} \]
- Substitute \( y \) into the fencing equation:
\[ P = 2 \left( \frac{125,000}{x} \right) + x \]
- To find the dimensions that minimize the fencing, take the derivative of \( P \) with respect to \( x \), set it to zero, and solve for \( x \).
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