Question

ricks lumberyard has 480 yd of fencing with which to enclose a rectangular area. if the enclosed area is x yards long, express its area as a function of its length. a(x)=\square

Explanation:

Step1: Recall the perimeter formula for a rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2(\text{length} + \text{width}) \). We know the perimeter \( P = 480 \) yards and the length is \( x \) yards. Let the width be \( w \) yards. So, \( 480=2(x + w) \).

Step2: Solve for the width \( w \)

First, divide both sides of the perimeter equation by 2: \( \frac{480}{2}=x + w \), which simplifies to \( 240=x + w \). Then, solve for \( w \) by subtracting \( x \) from both sides: \( w = 240 - x \).

Step3: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A=\text{length}\times\text{width} \). We know the length is \( x \) and the width is \( 240 - x \), so the area \( A(x)=x\times(240 - x) \).

Step4: Expand the expression (optional, but to write in standard form)

Using the distributive property, \( A(x)=240x - x^{2}=-x^{2}+240x \).

Answer:

\( A(x)=-x^{2} + 240x \) (or \( A(x)=x(240 - x) \))