Question

as part of a recreational waterfront grant, the city council plans to enclose a rectangular area along the waterfront of a lake and create a park and swimming area. the budget calls for the purchase of 8,000 feet of fencing. (note: there is no fencing along the lake.) let x represent the length of one of the two equal sides that are perpendicular to the water. complete parts (d), (e), and (f). (d) determine the value of x for which the area, a(x), of this rectangular site is a maximum. x = \square (simplif integer or a simplified fraction.) feet square feet cubic feet

Explanation:

Step1: Define Variables and Fencing Length

Let \( x \) be the length of the sides perpendicular to the water, and \( y \) be the length parallel to the water. The total fencing is \( 2x + y = 8000 \), so \( y = 8000 - 2x \).

Step2: Area Function

The area \( A(x) = x \cdot y = x(8000 - 2x) = -2x^2 + 8000x \). This is a quadratic function with \( a = -2 \), \( b = 8000 \).

Step3: Find Maximum of Quadratic

For a quadratic \( ax^2 + bx + c \), the vertex (maximum, since \( a < 0 \)) is at \( x = -\frac{b}{2a} \). Substituting \( a = -2 \), \( b = 8000 \):
\( x = -\frac{8000}{2(-2)} = \frac{8000}{4} = 2000 \).

Answer:

2000