Problem 146 - Algebra and Trigonometry

Question

A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.

Step-by-Step Solution

Answer

The width of the parking lot is 12 yards and the length is 15 yards.

Step 1: Define the Variables

Let the width of the parking lot be denoted by \( w \) (in yards), and the length be denoted by \( l \) (in yards). According to the problem we know that \( l = w + 3 \), and \( l \cdot w = 180 \)

Step 2: Substitute and Solve

Substitute \( l = w + 3 \) into the equation for the area, \( l \cdot w = 180 \). This gives the equation \( (w + 3) \cdot w = 180 \), which simplifies to \( w^2 +3w -180 = 0 \). Solve this equation for \( w \). This equation is quadratic and can be factored to \( (w -12)(w + 15) = 0 \). So we get two roots \( w = 12 \) and \( w = -15 \)

Step 3: Verify the Solution

The width can't be negative, so \( w = -15 \) is not acceptable. Therefore, the width of the parking lot \( w = 12 \) yards. Substitute \( w = 12 \) into \( l = w + 3 \), we get \( l = 12 + 3 = 15 \) yards.