Forming a Pen We want to form a rectangular pen of area 100 square feet. One side of the pen is to be formed by an existing building and the other three sides by a fence (see Figure 2.106). Let W be the length, in feet, of the sides of the rectangle perpendicular to the building, and let L be the length, in feet, of the other side.

e. Make a graph of F versus W.

Here's the graph of the fencing required, \( F \), versus the width, \( W \) of the pen. The curve shows how the amount of fencing changes with different widths while maintaining an area of 100 square feet.

You might notice a minimum point on the graph. This is where the fencing requirement is minimized, and it occurs when \( W = \sqrt{50} \). This suggests an optimal width for minimizing the amount of fencing needed.

Do you have any questions or would you like further details?

Here are 5 related questions to explore further:

1. What is the optimal width \( W \) that minimizes the amount of fencing?
2. How does changing the area of the pen affect the graph?
3. What is the total length of fencing required at the optimal width?
4. How would the fencing equation change if two sides of the pen were against the building?
5. What happens to the fencing requirement as \( W \) approaches zero?

**Tip:** The optimal point on the graph can be found using calculus by setting the derivative of \( F \) with respect to \( W \) to zero.

We want to form a rectangular pen of area 100 square feet. One side of the pen is to be formed by an existing building and the other three sides by a fence. Let W be the length, in feet, of the sides of the rectangle perpendicular to the building, and let L be the length, in feet, of the other side. Make a graph of F versus W.

Learn how to optimize the amount of fencing required to form a rectangular pen with a fixed area of 100 square feet, where one side is formed by an existing building. This problem involves key algebraic and geometric concepts and demonstrates how to minimize fencing using optimization techniques. Suitable for students in Grades 10-12.

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