Problem 97 - Algebra and Trigonometry

Question

A factory is to be built on a lot measuring 180 \(\mathrm{ft}\) by 240 \(\mathrm{ft}\) . A local building code specifies that a lawn of uniform width and equal in area to the factory must surround the factory. What must the width of this lawn be, and what are the dimensions of the factory?

Step-by-Step Solution

Answer

The width of the lawn is 30 feet. The factory dimensions are 120 feet by 180 feet.

Step 1: Determine Total Area of the Lot

The total area of the lot is given by multiplying its length and width: \( 180 \text{ ft} \times 240 \text{ ft} = 43200 \text{ ft}^2 \).

Step 2: Define Areas Algebraically

Let \( x \) be the width of the lawn. The dimensions of the factory will be \( (180 - 2x) \) feet by \( (240 - 2x) \) feet due to the uniform surrounding lawn. Hence, the area of the factory is \((180 - 2x)(240 - 2x)\) square feet.

Step 3: Equation for Equal Areas of Factory and Lawn

Since the lawn area and the factory area are equal, we set up the equation: \((180 - 2x)(240 - 2x) = \frac{43200}{2}\). This simplifies to the equation \((180 - 2x)(240 - 2x) = 21600\).

Step 4: Expand and Simplify the Equation

Expand \((180 - 2x)(240 - 2x)\): \[(180 \cdot 240) - (180 \cdot 2x) - (240 \cdot 2x) + (4x^2) = 21600\] \[43200 - 360x - 480x + 4x^2 = 21600\]Simplify to get:\[4x^2 - 840x + 43200 = 21600\].

Step 5: Solve the Quadratic Equation

Subtract 21600 from both sides:\[4x^2 - 840x + 21600 = 0\].Divide the entire equation by 4 for simplicity:\[x^2 - 210x + 5400 = 0\]. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1, b=-210, c=5400\).

Step 6: Calculate the Roots

Calculate the discriminant:\[b^2 - 4ac = (-210)^2 - 4 \cdot 1 \cdot 5400 = 44100 - 21600 = 22500\].Take the square root and solve:\[x = \frac{210 \pm \sqrt{22500}}{2}\].\[x = \frac{210 \pm 150}{2}\].The solutions are \(x = 180\) (not feasible) and \(x = 30\).

Step 7: Determine Factory Dimensions

With \(x = 30\), substitute back to find factory dimensions:Length: \(180 - 2(30) = 120\) feetWidth: \(240 - 2(30) = 180\) feet.

Key Concepts

Area Calculation

Calculating the area of a space is fundamental in determining how much space is available or needed for a particular purpose. In the given problem, we need to calculate the area of both a lot and a factory. The total area of a lot is straightforward; it's the product of its length and width.For the lot, we have:

• Length: 180 feet
• Width: 240 feet

So, the area is:\[180 \text{ ft} \times 240 \text{ ft} = 43200 \text{ ft}^2\]Next, we calculate the area of the factory, which is surrounded by a lawn, meaning there's a portion of the lot that is not the factory. What makes this problem interesting is that the lawn's area is equal to the area of the factory.The need for uniform width means the factory's length and width dimensions are reduced on each side by the same amount, twice the width of the lawn. Hence, finding how much area is allocated to each purpose (lawn vs. factory) allows for solving practical problems like knowing the dimensions of the factory.

Algebraic Expressions

Algebraic expressions allow us to generalize mathematical problems, turning a word problem into a solvable equation. For this problem, we start by letting the width of the lawn be \( x \).Because the lawn surrounds the factory uniformly, the factory's length and width are both reduced by \( 2x \) (since the lawn extends on both sides for both the length and the width).Thus, the expressions for the factory's dimensions become:

• Length: \( 180 - 2x \)
• Width: \( 240 - 2x \)

The area of the factory is then given by the product of these dimensions: \((180 - 2x)(240 - 2x)\). This expression helps us set the equation where the area of the lawn is equal to the area of the factory since half of the lot's total area would belong to each component:\[(180 - 2x)(240 - 2x) = 21600\]Using algebraic expressions and such equations, we can simplify and solve for \( x \), the lawn's width, thereby making these expressions a powerful tool in determining all necessary measurements and verifying their correctness.

Problem Solving Steps

Problem-solving in mathematics often involves a series of logical steps to transform a real-world problem into solvable equations. Initially, it’s crucial to comprehend the problem statement fully. Here, we are tasked with determining both the width of a lawn surrounding a factory and the dimensions of the factory itself.To solve this:1. **Understand Total and Shared Areas**: The total area of the lot is known. Recognize that the lawn area equals the factory area.2. **Set Up the Equation**: Use algebraic expressions to model the problem’s requirements. The critical step is creating the equation \((180 - 2x)(240 - 2x) = 21600\), representing equal areas.3. **Solve the Quadratic Equation**: - Expand the expression to a quadratic form: \(4x^2 - 840x + 21600 = 0\) - Factor or use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find \(x\).4. **Apply Results**: With \(x = 30\), calculate dimensions of the factory: length is \(120\) feet, width is \(180\) feet.This structured method allows us to systematically work through potentially complex mathematical problems by breaking them down into manageable parts.