Problem 23 - Precalculus Essentials

Question

As in Exercise \(21,\) you have 800 feet of fencing to enclose a rectangular field. However, one side of the field lies along a canal and requires no fencing. 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 dimension \(x\) is: \(A = 800x - 2x^2\)

Step 1: Define Variables

Firstly, define \(x\) as the dimension which is fenced on both sides (the width), and \(y\) as the length of the rectangle which is fenced only on one side.

Step 2: Equation for Total Fencing Available

Given that a total of 800 feet of fencing are available, and only three sides of the rectangle are being fenced, we have that \(2x + y = 800\). From this equation, we can express \(y\) in terms of \(x\) to get \(y = 800 - 2x\).

Step 3: Express Area in Terms of Width (x)

The area of a rectangle is calculated by multiplying its length by its width, \(A = x*y\). Replacing \(y\) with the expression obtained in Step 2, we get \(A = x*(800 - 2x)\), which simplifies to \(A = 800x - 2x^2\). This function represents the area of the field as a function of its dimension \(x\).

Key Concepts

Area of a Rectangle

The area of a rectangle is a fundamental concept in geometry, representing the amount of space inside the rectangle. To calculate it, simply multiply its length and width.
Think of it like covering the rectangle with 1x1 square tiles; the area tells you how many tiles you'll need.
It's expressed as \(A = l \times w\), where \(l\) is the length, and \(w\) is the width.

• When calculating, ensure both dimensions are in the same units.
• The unit of area can be square units, like square feet or square meters.

Understanding how area changes with different dimensions is crucial. In optimization problems, like the rectangular fencing problem, we often explore how to maximize or minimize this area by manipulating dimensions.
By analyzing the relationships between length and width, and sometimes converting one in terms of the other, we can find the best dimensions for specific conditions.

Rectangular Fencing Problem

The rectangular fencing problem is a classic question in optimization. It involves figuring out the best way to use a given amount of fencing to enclose a rectangular area.
In our scenario, we are given 800 feet of fencing, but only need to enclose three sides of a rectangle, with one side lying along a canal.
This changes the usual approach:

• Two short or parallel sides: These are opposite each other and are both fenced.
• One long or unopposed side: Mimicking the canal side and requiring no fencing.

The task is to express the area as a function of one dimension, \(x\), considering the relation between both sides. If \(x\) represents the sides needing fencing, then the opposite side is calculated using \(y = 800 - 2x\). This provides a method to assess how changes in dimension \(x\) will impact the enclosed area.

Functions and Equations

Functions and equations play a critical role in problems involving optimization in geometry. Here, they help express relationships between dimensions and outcomes, like area.
In the fencing problem, we determine an equation from the physical setup, such as \(2x + y = 800\), representing how we allocate the 800 feet of fencing.
By isolating one variable, \(y\), in terms of the other, \(x\), we derive \(y = 800 - 2x\). This simplifies further calculations.

• Function of Area: Using equations, we express the area \(A\) as \(A = x(800 - 2x)\).
• Analyzing this function helps us understand how changing \(x\) affects \(A\).

Optimizing the function is about finding the best \(x\) that will maximize the area \(A\). This involves using techniques from calculus or simple logical reasoning, exploring how the graph of the function behaves around its maximum point.