Dimensions of a Parking Lot. A parking lot has a rectangular area of 40,000 yd². The length is 200 yd more than twice the width. Find the dimensions of the lot.

Verified step by step guidance

1. Let the width of the parking lot be represented by \(w\) (in yards).
2. Express the length \(L\) in terms of the width \(w\) using the given relationship: \(L = 2w + 200\).
3. Write the equation for the area of the rectangle using the formula \(\text{Area} = \text{length} \times \text{width}\), which gives \$40000 = w \times (2w + 200)$.
4. Expand and simplify the equation to form a quadratic equation: \$40000 = 2w^2 + 200w$.
5. Rearrange the quadratic equation to standard form: \$2w^2 + 200w - 40000 = 0\(, then solve for \)w$ using the quadratic formula or factoring.

Key Concepts

Formulating Algebraic Equations from Word Problems

This involves translating the given verbal information into mathematical expressions. For example, representing the length as '200 yd more than twice the width' translates to L = 2W + 200. Understanding how to set up these relationships is essential for solving the problem.

Area of a Rectangle

The area of a rectangle is found by multiplying its length by its width (Area = Length × Width). Knowing this formula allows you to set up an equation using the given area and the expressions for length and width to find unknown dimensions.

Solving Quadratic Equations

After substituting the expressions for length and width into the area formula, you get a quadratic equation. Solving this equation using factoring, completing the square, or the quadratic formula helps find the possible values for the width, and subsequently the length.