You Have 332 Feet Of Fencing To Enclose

you Have 332 Feet Of Fencing To Enclose

Identify the core question: The task is to find the maximum area of a rectangular region enclosed by 332 feet of fencing.

Restate the problem: Given a fixed amount of fencing, what dimensions of a rectangle will maximize the enclosed area? The problem involves optimizing a rectangle’s area under a perimeter constraint.

Paper For Above instruction

To determine the maximum area that can be enclosed with 332 feet of fencing, we first model the problem mathematically. Let the length of the rectangle be represented by \( x \) and the width by \( y \). The total fencing corresponds to the perimeter \( P \), expressed as:

\( P = 2(x + y) = 332 \)

From this, we can derive \( y \) in terms of \( x \):

\( y = \frac{332}{2} - x = 166 - \frac{x}{2} \)

The area \( A \) of the rectangle is given by:

\( A(x) = x \times y = x \left( 166 - \frac{x}{2} \right) = 166x - \frac{x^2}{2} \)

To find the maximum area, differentiate \( A(x) \) with respect to \( x \):

\( A'(x) = 166 - x \)

Set the derivative equal to zero to find the critical point:

\( 166 - x = 0 \Rightarrow x = 166 \)

Substitute \( x = 166 \) back into the expression for \( y \):

\( y = 166 - \frac{166}{2} = 166 - 83 = 83 \)

The dimensions that maximize the enclosed area are 166 feet in length and 83 feet in width. The maximum area is then:

\( A_{max} = 166 \times 83 = 13,778 \text{ square feet} \)

Therefore, the maximum enclosed area with 332 feet of fencing is 13,778 square feet, achieved when the rectangle measures 166 feet by 83 feet.

References

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