Question

paul has 3600 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area? a rectangle that maximizes the enclosed area has a length of \boxed{} yards and a width of \boxed{} yards.

Explanation:

Step1: Define variables and perimeter formula

Let the length of the rectangle be \( l \) and the width be \( w \). The perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \). We know that the total fencing (perimeter) is 3600 yards, so \( 2l + 2w = 3600 \). We can simplify this equation by dividing both sides by 2: \( l + w = 1800 \), and then express \( l \) in terms of \( w \): \( l = 1800 - w \).

Step2: Define the area formula

The area \( A \) of a rectangle is given by \( A = l \times w \). Substitute \( l = 1800 - w \) into the area formula: \( A(w) = (1800 - w)w = 1800w - w^2 \).

Step3: Find the maximum of the quadratic function

The area function \( A(w) = -w^2 + 1800w \) is a quadratic function in the form \( f(x) = ax^2 + bx + c \), where \( a = -1 \), \( b = 1800 \), and \( c = 0 \). For a quadratic function, the vertex (which gives the maximum or minimum value) occurs at \( w = -\frac{b}{2a} \). Since \( a = -1 < 0 \), the parabola opens downward, so the vertex is the maximum point.

Calculate \( w \):
\[
w = -\frac{1800}{2(-1)} = \frac{1800}{2} = 900
\]

Step4: Find the length

Substitute \( w = 900 \) back into the equation for \( l \): \( l = 1800 - 900 = 900 \).

Answer:

A rectangle that maximizes the enclosed area has a length of \(\boldsymbol{900}\) yards and a width of \(\boldsymbol{900}\) yards. (Note: The maximum area would be \( A = 900 \times 900 = 810000 \) square yards, but the question here asks for the dimensions.)