Problem 22 - Precalculus Essentials

Question

You have 600 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 rectangle, \(A\), can be expressed as a function of one of its dimensions, say the length \(x\), and is given by \(A = 300x - x^2\).

Step 1: Express the Breadth in terms of the Length

Given that the total length of fence, which represents the perimeter (P) of the field, is 600 feet, we can express this as \(P = 2(length + breadth)\). Let the dimension \(x\) represent the length of the rectangle. Then, we can express the breadth (b) in terms of the length (x) using the formula of the perimeter. Hence, \(breadth = (P/2) - length = (600/2) - x = 300 - x\)

Step 2: Express the Area as a function of the Length

The area (A) of a rectangle is given by the product of its length and breadth. Now that we have the breadth in terms of length, we can substitute it in the area formula. Thus, \(A = length * breadth = x * (300 - x) = 300x - x^2\)