Problem 67 - Calculus Early Transcendentals

Question

A rectangular plot of land is to be fenced off so that the area enclosed will be \(400 \mathrm{ft}^{2}\). Let \(L\) be the length of fencing needed and \(x\) the length of one side of the rectangle. Show that \(L=2 x+800 / x\) for \(x>0,\) and sketch the graph of \(L\) versus \(x\) for \(x>0 .\)

Step-by-Step Solution

Answer

The formula is \(L = 2x + \frac{800}{x}\). The graph shows a 'U'-shaped curve for \(x > 0\).

Step 1: Define the Variables and Given Condition

We are given a rectangular plot with an area of \(400 \text{ ft}^2\). Let \(x\) be the length of one side of the rectangle. The area of the rectangle is given by the formula \(\text{Area} = x \times y = 400 \text{ ft}^2\), where \(y\) is the length of the other side.

Step 2: Express the Other Side of the Rectangle in Terms of x

From the given area \(x \times y = 400\), we can solve for \(y\) in terms of \(x\): \[ y = \frac{400}{x} \]

Step 3: Write the Formula for the Perimeter (Fencing Needed)

The length of fencing needed, \(L\), is the perimeter of the rectangle. The perimeter of a rectangle is given by \(2x + 2y\). Substitute \(y = \frac{400}{x}\) into the perimeter formula: \[ L = 2x + 2\left(\frac{400}{x}\right) \]

Step 4: Simplify the Expression for L

Simplify the expression for the perimeter: \[ L = 2x + \frac{800}{x} \] Thus, the formula for the length of fencing needed is \(L = 2x + \frac{800}{x}\), for \(x > 0\).

Step 5: Sketch the Graph of L versus x

To sketch the graph of \(L = 2x + \frac{800}{x}\), note the behavior as \(x\) changes:- As \(x\) approaches zero, \(\frac{800}{x}\) becomes very large, making \(L\) very large.- As \(x\) increases, \(\frac{800}{x}\) decreases, and \(L\) approaches a minimum point before increasing due to the \(2x\) term.Plot several points using values of \(x\) to capture this behavior and connect them smoothly.

Key Concepts

Perimeter Formulas

Understanding perimeter formulas is essential when dealing with optimization problems like this one. The perimeter of a rectangle is the total distance around the edge of the shape. For any rectangle, the formula for calculating the perimeter is given by \[ P = 2 imes ( ext{length} + ext{width}) \] Where the length and width are the dimensions of the rectangle. In our specific problem, these dimensions were represented by the variables \(x\) and \(y\) with the additional constraint of a fixed area of \(400 \text{ ft}^2\).
By recognizing these relationships, we were able to derive the expression for \(L\), the length of fencing required, as a function of \(x\). This derived formula \(L = 2x + \frac{800}{x}\) allows us to determine the necessary length of fencing based on different values of \(x\). Understanding perimeter formulas in this context illustrates how we systematically balance and adjust these dimensions to optimize for certain conditions, such as minimization of material used.

Rectangular Area

The concept of rectangular area is foundational in solving optimization problems involving fenced regions. The area of a rectangle is determined by the formula: \[ ext{Area} = ext{length} imes ext{width} \] In this exercise, the area is a constant \(400 \text{ ft}^2\).
This means regardless of our choice of one dimension, say \(x\), the other dimension \(y\) must adjust to maintain this constant area. By manipulating the formula to express \(y\) in terms of \(x\) as \(y = \frac{400}{x}\), we create a relationship that helps us explore different dimension pairings while keeping the area fixed.
Understanding how to manipulate these relationships is key in optimization, as it allows us to explore various scenarios within given constraints, such as minimizing or maximizing a related value like the perimeter.

Graphing Functions

Graphing functions is a practical way to visually analyze relationships and behaviors in mathematical problems. In this exercise, we sought to graph the function describing the length of fencing needed, \(L = 2x + \frac{800}{x}\). Observing graph behavior provides insights into how the length of fencing changes as the variable \(x\) changes.
When graphing, key features to note include:

• How \(L\) increases sharply as \(x\) approaches zero, due to \(\frac{800}{x}\) becoming a large value.
• How \(L\) decreases to a minimum point before rising again as \(x\) grows larger due to the \(2x\) term.

Plotting several strategic points can help visualize this behavior, showing both extremes and more moderate values, connecting the dots into a smooth curve. This graph helps in understanding where the minimal or optimal fence length is achieved, painting a clear picture of the mathematical relationship between \(x\) and \(L\).
Graphing functions, thus, is an invaluable tool in optimization problems, aiding in decision-making by presenting a clear visual representation of mathematical relationships.