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) = 8000000 square feet
(simplify your answer. type an integer or a simplified fraction.)
(f) what are the dimensions of the maximum enclosed area?
the length (dimension perpendicular to the water) of the maximum enclosed area is \\(\square\\) \\(\boldsymbol{\
abla}\\), and the width (dimension parallel to the water) of the maximum enclosed area is \\(\square\\) \\(\boldsymbol{\
abla}\\)

Explanation:

Step1: Recall the fencing and area setup

Let \( x \) be the length perpendicular to the water (so there are two sides of length \( x \)), and \( y \) be the length parallel to the water. The total fencing used is \( 2x + y=8000 \), so \( y = 8000 - 2x \). The area \( A(x)=x\times y=x(8000 - 2x)=- 2x^{2}+8000x \).

Step2: Find the value of \( x \) for maximum area (from part d, \( x = 2000 \))

We know from part (d) that the value of \( x \) (length perpendicular to water) that maximizes the area is \( x = 2000 \) feet.

Step3: Find the value of \( y \) (length parallel to water)

Substitute \( x = 2000 \) into the equation for \( y \): \( y=8000 - 2x \).
\( y=8000-2\times2000=8000 - 4000 = 4000 \) feet.

So the length perpendicular to the water is \( 2000 \) feet and the length parallel to the water is \( 4000 \) feet.

Answer:

The length (dimension perpendicular to the water) of the maximum enclosed area is 2000 feet, and the width (dimension parallel to the water) of the maximum enclosed area is 4000 feet.