Problem 145 - College Algebra Essentials

Question

The length of a rectangular sign is 3 feet longer than the width. If the sign's area is 54 square feet, find its length and width.

Step-by-Step Solution

Answer

The width of the rectangular sign is 6 feet and the length is 9 feet.

Step 1: Let Length and Width be represented by Variables

Let’s say the width of the rectangle is \(x\). Therefore, the length has to be \(x + 3\) because the length is 3 feet longer than the width.

Step 2: Set Up the Area Equation

The area of a rectangle is the product of its length and width. So, \(x(x + 3) = 54\)

Step 3: Solve the Equation

Rewrite the equation to standard quadratic form. We get \(x^2 + 3x - 54 = 0\), which can be factored into \((x - 6)(x + 9) = 0\). Solving for \(x\) gives \(x = 6\) and \(x = -9\). Since the width of a rectangle cannot be negative, \(x = 6\) is selected as the solution.

Step 4: Determine the Length

Now substitute \(x = 6\) into the equation \(x + 3\), we get the length to be 9 ft