Problem 21 - Precalculus Essentials

Question

You have 800 feet of fencing to enclose a rectangular field. Express the area of the field, \(A\), as a function of one of its dimensions, \(x\).

Step-by-Step Solution

Answer

The area of the field as a function of one of its dimensions, \(x\), can be expressed by the equation \(A = 400x - x^2\).

Step 1: Write Down the Equation for the Area of a Rectangle

For a rectangle, the area is given by the length times the width. If one side of the rectangle is \(x\), let's say the length, the width is determined by the remaining amount of fencing left after removing the two lengths of \(x\), i.e., the width is equal to \((800 - 2x)/2\), remember the perimeter of the rectangle is 800, which equals \(2*length + 2*width\). So the area, \(A\), of the field is therefore \(x \times ((800 - 2x)/2)\).

Step 2: Simplify the Equation for the Area

The equation in step 1 is correct for finding the field's area in terms of \(x\), but it would be advantageous to simplify it. To simplify the expression for the area, distribute \(x\) across the terms in the brackets. This results in \(A = x \times 800/2 - x \times 2x/2 = 400x - x^2\).

Step 3: Write Down the Function of the Area in Terms of \(x\)

The equation \(A = 400x - x^2\) represents the area of the rectangular field, \(A\), as a function of the dimension \(x\). Since \(A\) and \(x\) are both variables representing the area and dimension of the field respectively, the equation shows how the shape of the field (changing \(x\)) impacts the size of the area enclosed.