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 10,000 feet of fencing. (note: there is no fencing along the lake.) complete parts a through c below.
b. write an expression that represents the width (side opposite the water) in terms of x.
an expression that represents the width in terms of x is 10000 - 2x. (simplify your answer)
c. write an equation that expresses the area a of this rectangular site as a quadratic function of x.
a = \square (simplify your answer)

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the product of its length and width, i.e., \( A=\text{length}\times\text{width} \).

Step2: Identify the length and width in terms of \( x \)

From part (b), we know that if we let the length parallel to the water be \( x \) (assuming \( x \) is the length of the side parallel to the lake), then the width (side opposite the water) is \( 10000 - 2x \).

Step3: Substitute the length and width into the area formula

Substitute length \( = x \) and width \( = 10000 - 2x \) into the area formula:
\( A = x\times(10000 - 2x) \)

Step4: Simplify the expression

Using the distributive property \( a(b + c)=ab+ac \) (here \( a = x \), \( b = 10000 \), \( c=-2x \)):
\( A=10000x-2x^2=-2x^2 + 10000x \)

Answer:

\( A = x(10000 - 2x) = -2x^2 + 10000x \)