Question

1. (25 points) a rectangular pen for a pet is under construction using 100 feet of fencing. (a) express the area a of the rectangular pen as a function of x, where x is the length of the pen. (b) for what value of x is the area the largest? (c) what is the maximum area?

Explanation:

Step1: Find the width in terms of \( x \)

The perimeter of a rectangle is \( P = 2(\text{length} + \text{width}) \). Given the perimeter \( P = 100 \) feet and length \( x \), we have \( 100 = 2(x + \text{width}) \). Solving for width: divide both sides by 2, \( 50 = x + \text{width} \), so width \( = 50 - x \).

Step2: Express area as a function of \( x \)

The area of a rectangle is \( A = \text{length} \times \text{width} \). Substituting length \( x \) and width \( 50 - x \), we get \( A(x) = x(50 - x) = -x^2 + 50x \).

Step3: Find \( x \) for maximum area (vertex of parabola)

For a quadratic function \( y = ax^2 + bx + c \), the vertex (maximum for \( a < 0 \)) is at \( x = -\frac{b}{2a} \). Here, \( a = -1 \), \( b = 50 \), so \( x = -\frac{50}{2(-1)} = 25 \).

Step4: Calculate maximum area

Substitute \( x = 25 \) into \( A(x) \): \( A(25) = 25(50 - 25) = 25 \times 25 = 625 \) square feet.

Answer:

(a) \( A(x) = -x^2 + 50x \)
(b) \( x = 25 \) feet
(c) Maximum area is \( 625 \) square feet