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 = 2000 feet (simplify your answer. type an integer or a simplified fraction.) (e) what is the maximum area that can be enclosed? a(x) = \square (simplify your answer. type an integer or a simplified fraction.) options: feet, square feet, cubic feet

Explanation:

Step1: Define the length parallel to the lake

Let the length of the side parallel to the lake be \( y \). The total fencing used is \( 2x + y = 8000 \), so \( y = 8000 - 2x \).

Step2: Define the area function

The area \( A(x) = x \times y = x(8000 - 2x)= - 2x^{2}+8000x \).

Step3: Find the vertex of the quadratic function

For a quadratic function \( ax^{2}+bx + c \), the x - coordinate of the vertex is \( x =-\frac{b}{2a} \). Here, \( a=-2 \), \( b = 8000 \), so \( x =-\frac{8000}{2\times(-2)}=2000 \) (which matches the given value in part (d)).

Step4: Calculate the maximum area

Substitute \( x = 2000 \) into the area function. First, find \( y=8000 - 2\times2000=4000 \). Then the area \( A(2000)=2000\times4000 = 8000000 \) square feet. Or using the quadratic formula: \( A(2000)=-2\times(2000)^{2}+8000\times2000=- 8000000 + 16000000=8000000 \) square feet.

Answer:

8000000 square feet