To address the problem (assumed to be the example mentioned in the system prompt), here's the step-by-step solution:

Problem Statement

A farmer has 100 meters of fencing to enclose a rectangular pasture adjacent to a river (no fencing needed along the river). Maximize the area of the pasture.

Step 1: Define Variables

Let:

• ( W = ) width of the pasture (perpendicular to the river, 2 sides needed).
• ( L = ) length of the pasture (parallel to the river, 1 side needed).

Step 2: Fencing Constraint

Total fencing used: ( L + 2W = 100 ) meters.
Express ( L ) in terms of ( W ):
( L = 100 - 2W ).

Step 3: Area Function

Area ( A = L \times W ). Substitute ( L ):
( A(W) = (100 - 2W)W = 100W - 2W^2 ).

Step 4: Maximize the Quadratic Function

The area function is a downward-opening parabola (( a = -2 < 0 )), so its maximum occurs at the vertex. For ( ax^2 + bx + c ), the vertex is at ( x = -\frac{b}{2a} ).

Here, ( a = -2 ), ( b = 100 ):
( W = -\frac{100}{2(-2)} = 25 ) meters.

Step 5: Compute Optimal Length and Area

( L = 100 - 2(25) = 50 ) meters.
Maximum area: ( A = 50 \times 25 = 1250 ) square meters.

Final Answer: The maximum area is (\boxed{1250}) square meters.

(\boxed{1250})

作者声明:本文包含人工智能生成内容。