Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

A square vegetable garden is to be tilled and then enclosed with a fence. If the fence costs \(\$ 1\) per foot and the cost of preparing the soil is \(\$ 0.50\) per \(\mathrm{ft}^{2}\), determine the size of the garden that can be enclosed for \(\$ 120\).

Step-by-Step Solution

Answer

The side length of the garden can be 12 feet.

Step 1: Define Variables

Let the side length of the square garden be \(x\) feet.

Step 2: Cost of the Fence

The perimeter of the square garden is \(4x\) feet. The cost of the fence is therefore \(4x \times 1 = 4x\) dollars.

Step 3: Cost of Preparing the Soil

The area of the garden is \(x^2\) square feet. The cost to prepare the soil is \(x^2 \times 0.50 = 0.5x^2\) dollars.

Step 4: Total Cost Equation

The total cost includes both the fence and soil preparation, which is given by the equation: \(4x + 0.5x^2 = 120\).

Step 5: Solve the Quadratic Equation

Rearrange the equation \(0.5x^2 + 4x - 120 = 0\). To simplify, multiply the entire equation by 2: \(x^2 + 8x - 240 = 0\).

Step 6: Use the Quadratic Formula

Solve for \(x\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 8\), and \(c = -240\).

Step 7: Calculate the Discriminant

The discriminant \(b^2 - 4ac = 8^2 - 4 \times 1 \times (-240) = 64 + 960 = 1024\).

Step 8: Solve for x

Substitute the discriminant back into the quadratic formula: \(x = \frac{-8 \pm \sqrt{1024}}{2}\). Since the square root of 1024 is 32, this becomes \(x = \frac{-8 \pm 32}{2}\).

Step 9: Select Practical Solution for x

Solving \(x = \frac{-8 + 32}{2}\) yields \(x = 12\). The solution \(x = \frac{-8 - 32}{2}\) is not practical, as it gives a negative value.

Key Concepts

Cost Optimization

Cost optimization is all about making smart decisions to get the most value for your money. In our square garden problem, we need to determine the optimal size of the garden so that the total cost does not exceed $120. This total cost includes the expenses for fencing and soil preparation.

By carefully balancing these costs, like deciding on the amount of fencing needed and the area to be prepared, we aim to minimize expenses while achieving the desired outcome. The concept of cost optimization here is applied by setting up an equation that represents the total cost in terms of the garden's side length, and then solving it to find the most economical size of the garden that stays within the budget.

This process helps us understand how mathematical tools, like quadratic equations, can be used to optimize costs in practical scenarios.

Perimeter and Area

Understanding the perimeter and area is crucial in solving the square garden problem, as these measurements directly influence cost calculations.

The perimeter of a square is calculated as four times the side length (\(4x\)). This measurement helps us determine the total length of fencing needed. On the other hand, the area is the square of the side length (\(x^2\)), which is necessary to compute the cost of soil preparation.

Knowing the formulas allows us to set up equations based on actual costs:

• Fence cost is based on the perimeter: \(4x \times \text{Cost per foot}\).
• Soil preparation cost is based on the area: \(x^2 \times \text{Cost per square foot}\).

Overall, mastering these concepts helps in calculating how much material is needed for the project, ultimately influencing the cost efficiency.

Square Garden Problem

In addressing the square garden problem, we deal with both a practical and a mathematical challenge. We need to enclose a garden with fencing and prepare its soil, all within a fixed budget.

Initially, we define the problem using a variable (\(x\)) to represent the side length of the square garden. The interplay between perimeter and area becomes key in expressing the total cost in terms of this variable. This leads to forming a quadratic equation:

• Give insight on costs by explaining how the perimeter (fence) and area (soil preparation) add up to a total budget.
• Transform the challenge to a problem of solving the equation, allowing prediction of the perfect dimension for the garden.

Ultimately, by solving this quadratic equation using the quadratic formula, we find the practical side length that fits within the budget. In this case, that solution is a garden with sides of 12 feet, which provides a successful application of the quadratic formula to a real-world cost optimization scenario.