Problem 51 - College Algebra

Question

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 \(\mathrm{ft}^{2}\) . Solve the quadratic equation to find the length and width.

Step-by-Step Solution

Answer

The width of the garden is 7 feet and the length is 17 feet.

Step 1: Define the Variables

Let the width of the garden be \( x \) feet. Then, according to the problem, the length would be \( x + 10 \) feet since the length is 10 feet longer than the width.

Step 2: Set Up the Area Equation

The area of the garden is given by the formula for the area of a rectangle, which is length times width. So, we set up the equation: \( x(x + 10) = 119 \).

Step 3: Expand and Rearrange the Equation

Expand the product and rearrange the equation to set it to standard quadratic form: \( x^2 + 10x - 119 = 0 \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 10 \), and \( c = -119 \) to find the width \( x \).

Step 5: Apply the Quadratic Formula

Calculating the discriminant: \( b^2 - 4ac = 10^2 - 4(1)(-119) = 100 + 476 = 576 \). The discriminant is positive, indicating two real solutions. Continue calculating: \( x = \frac{-10 \pm \sqrt{576}}{2} = \frac{-10 \pm 24}{2} \).

Step 6: Find the Solutions for x

The solutions are \( x = \frac{-10 + 24}{2} = \frac{14}{2} = 7 \), and \( x = \frac{-10 - 24}{2} = \frac{-34}{2} = -17 \). Since negative width isn't feasible, we have \( x = 7 \) feet.

Step 7: Determine Length and Confirm

Now, compute the length using \( x = 7 \) feet. Length = \( x + 10 = 17 \) feet. Confirm area: \( 7 \times 17 = 119 \ \text{ft}^2 \) (Correct calculation).

Key Concepts

Area of a Rectangle

The area of a rectangle is a basic concept in geometry that involves calculating the space contained within a rectangular shape. The formula for finding the area is simple: multiply the length by the width.

• If you have a rectangle with a length of \( L \) and a width of \( W \), then the area \( A \) is given by: \( A = L \times W \).
• This equation allows you to calculate how much surface the rectangle covers, which is especially handy in real-world applications like gardening, flooring, and painting.

In this exercise, the garden's length is 10 feet longer than its width, leading to a unique equation derived from the area formula: \( x(x + 10) = 119 \). Here, \( x \) represents the width, and \( x + 10 \) gives us the length. Understanding how to apply this formula helps bridge the gap between abstract mathematical concepts and practical applications.

Solving Quadratic Equations

Solving quadratic equations is a vital skill in algebra. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \).

• To solve it, you need to find the values of \( x \) that make the equation true.
• The solutions are known as the roots, and they can tell whether an equation has real, complex, or repeated roots.

In this exercise, we needed to rearrange the equation derived from the area to \( x^2 + 10x - 119 = 0 \), putting it in the standard quadratic form. Solving this will give us the dimensions of the garden. The process usually involves methods such as factoring, using the quadratic formula, or completing the square, depending on which is most convenient or possible.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It provides a straightforward way to find solutions by converting any quadratic equation into a general solution format.

• The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), used when the equation is in the standard form \( ax^2 + bx + c = 0 \).
• This exercise showed its application by solving \( x^2 + 10x - 119 = 0 \), where \( a = 1 \), \( b = 10 \), and \( c = -119 \).
• We computed the discriminant, \( b^2 - 4ac \), to ensure real solutions, finding it to be \( 576 \).

The positive discriminant indicated two real solutions, \( x = 7 \) and an impractical negative length. Hence, the quadratic formula guided us to find the feasible width and length for the garden, transforming the theoretical into a practical answer.