Problem 101 - Algebra and Trigonometry

Question

Width of a Lawn 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 lawn must be 60 ft wide, and the factory dimensions are 60 ft by 120 ft.

Step 1: Determine the Lot Area

The lot measures 180 ft by 240 ft, so the area of the lot is given by multiplying these dimensions: \( 180 \times 240 = 43200 \, \text{square feet} \).

Step 2: Define Variables for Dimensions

Let \( x \) represent the width of the lawn. The factory dimensions will then be \((180 - 2x)\) by \((240 - 2x)\) to account for the lawn on each side.

Step 3: Establish the Equation for Equal Areas

The area of the factory must equal the area of the lawn. Thus, we set up the equation: \( (180 - 2x)(240 - 2x) = x((240 \times 180) - (180 - 2x)(240 - 2x)) \).

Step 4: Simplify and Solve the Equation

First, expand the expression for the factory area: \( (180 - 2x)(240 - 2x) = 43200 - 840x + 4x^2 \). Since the lot area is \(43200\), the factory area is \(43200 - \text{Lawn Area}\). Solving for \(x^2 - 180x + 21600 = 0\) gives us \(x = 60\) or \(x = 120\).

Step 5: Determine Feasible Solution

Out of the solutions, \( x = 60 \) is feasible since it gives positive dimensions for the factory. With \(x = 60\), the factory dimensions are \(180 - 2(60) = 60\) and \(240 - 2(60) = 120\).

Step 6: Verify the Conditions

Verify if the area of the factory matches the area of the lawn. Both areas should be \(60 \times 120 = 7200 \text{ square feet}\), confirming the correct width.

Key Concepts

Area calculation

When solving algebra problems that involve geometric figures, understanding area calculation is crucial. In this exercise, we deal with a lot and need to calculate the area of both the lot and the lawn that surrounds a factory.
To start, determine the total area of the lot by multiplying its length by its width. If the lot dimensions are given as 180 feet by 240 feet, calculate the area with:

• Area of the lot: \[180\, \times\, 240 = 43200\, \text{square feet}\]

Once the lot's total area is known, this information helps in figuring out the area that the lawn and the factory will each occupy. This requires setting up an equation for equal areas, since according to the problem, the factory's area must be equal to the lawn's area.

Quadratic equation

The heart of this algebra problem is setting up a quadratic equation to determine the width of the lawn. Quadratic equations typically take the form:

• \[ax^2 + bx + c = 0\]

In this situation, we define the width of the lawn as \(x\), and the equation is formed by expressing the factory's area in terms of \(x\) and setting it equal to the lawn's area.
This particular problem leads to the equation:

• \[x^2 - 180x + 21600 = 0\]

Where solving for \(x\) gives you possible values for the width of the lawn. Factoring or using the quadratic formula helps find the feasible solution, which affects the factory and lawn assignment.

Geometric modeling

Geometric modeling involves visualizing the physical layout in mathematical terms. Imagine a rectangular lot with a factory at its center and a lawn surrounding it. The problem requires visual representation of:

• Factory: A rectangle inside the larger lot.
• Lawn: A strip of uniform width around the factory.

Considering the lawn's uniform width \(x\), the factory dimensions reduce by twice this width on each side, modeled thus:

• For the length, it becomes \(180-2x\).
• For the width, it's \(240-2x\).

Modeling in this manner allows setting up equations for solving the problem of equal area distribution and determining the feasible dimensions.

Dimension analysis

In dimension analysis, the goal is to scrutinize the relationships between different measurements to ensure logical coherence. Start by analyzing how changing one dimension affects others.
For this exercise, if \(x\) is the lawn's width:

• The factory's new length: \(180-2x\).
• The factory's new width: \(240-2x\).

Once these relationships are understood, ensure that substituting \(x = 60\) results in the dimension constraints of the problem, making both the length and width non-negative.
After concluding the analysis, verify that these dimensions correctly allow the factory and lawn to each take up half the total lot area, maintaining all stated conditions.