Problem 88 - Algebra and Trigonometry

Question

Fencing a Garden \(\quad\) A determined gardener has 120 \(\mathrm{ft}\) of deer- resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area enclosed to be at least 800 \(\mathrm{ft}^{2} .\) What range of values is possible for the length of her garden?

Step-by-Step Solution

Answer

The possible range for the length of the garden is from 20 ft to 40 ft.

Step 1: Define Variables

Let's define the variables first. Let \( L \) represent the length and \( W \) represent the width of the garden. Since it is a rectangle, the perimeter can be given by \( 2L + 2W = 120 \). Additionally, the area of the garden is given by \( L \times W \geq 800 \).

Step 2: Express Width in terms of Length

Use the perimeter equation to express \( W \) in terms of \( L \). From \( 2L + 2W = 120 \), divide by 2 to get \( L + W = 60 \). Hence, we can express \( W \) as \( W = 60 - L \).

Step 3: Substitute Width into Area Equation

Replace \( W \) in the area equation with the expression we found: \( L(60 - L) \geq 800 \). This simplifies to \( 60L - L^2 \geq 800 \).

Step 4: Rearrange to Form a Quadratic Equation

Rearrange the inequality \( 60L - L^2 \geq 800 \) to get \( -L^2 + 60L - 800 \geq 0 \). This can be rewritten as \( L^2 - 60L + 800 \leq 0 \).

Step 5: Solve Quadratic Equation

We need to solve \( L^2 - 60L + 800 = 0 \) to find critical points. Use the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1, b = -60, c = 800 \). This gives \( L = \frac{60 \pm \sqrt{3600 - 3200}}{2} = \frac{60 \pm \sqrt{400}}{2} \). Calculating further, \( L = \frac{60 \pm 20}{2} \), we find \( L = 40 \) or \( L = 20 \).

Step 6: Determine Range of Length Values

The quadratic inequality \( L^2 - 60L + 800 \leq 0 \) holds between the roots. Thus, the length \( L \) can range from 20 to 40, inclusive, to satisfy both the perimeter and area conditions.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is an essential concept for understanding the dimensions of a rectangular shape. It refers to the total distance around the outside of the rectangle. To determine the perimeter, one needs to add up the lengths of all four sides. Typically, if we denote the length of the rectangle as \( L \) and the width as \( W \), the formula for the perimeter \( P \) is:
\[P = 2L + 2W\]The perimeter is particularly useful when planning to enclose an area with materials such as fencing, as in this exercise.

• Perimeter informs the total needed fencing material for four sides.
• Breaking the perimeter equation down, you can better understand how every change in length or width affects the perimeter.
• For the given problem, it means working within a fixed perimeter of 120 feet to determine suitable dimensions for the rectangle.

Area of a Rectangle

The area of a rectangle is a fundamental concept used to find out how much space is enclosed within the rectangle. It quantifies the number of square units covered by the rectangle. The formula for calculating the area is straightforward:
\[A = L \times W\]Where \( A \) is the area, \( L \) is the length, and \( W \) is the width.

• The area is crucial when you want to determine usable space within a boundary, like a garden.
• In practical applications, such as the exercise, it helps in ensuring the space satisfies certain conditions, like a minimum area requirement.
• For this scenario, the gardener wants the area to be at least 800 square feet. This constraint works alongside the perimeter constraint to shape the possible dimensions of the garden.

Quadratic Formula

The quadratic formula is a vital tool in algebra, commonly used to solve equations that can be expressed in the form:
\[ax^2 + bx + c = 0\]Where \( a \), \( b \), and \( c \) are coefficients. The solution to these quadratic equations is given by the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows for finding the roots of the quadratic equation, which are the values of \( x \) that satisfy the equation.

• The formula works by providing critical points where the quadratic equation crosses the x-axis on a graph.
• In this problem, the quadratic formula helps identify the range of suitable lengths for the garden.
• By solving \( L^2 - 60L + 800 = 0 \) using the quadratic formula, the gardener finds the critical lengths 20 feet and 40 feet, giving insight into possible garden dimensions.