Problem 111 - College Algebra

Question

The length of a rectangular garden is 5 feet greater than the width. The area of the garden is 300 square feet. Find the length and the width.

Step-by-Step Solution

Answer

The width of the garden is 15 feet and the length is 20 feet.

Step 1: Define algebraic expressions

Let’s denote the width of the garden as \( w \) (in feet). Given the length of the garden is 5 feet greater than the width, so the length can be expressed as \( w + 5 \) feet. Also, it's known that the area of the garden, which is equal to length times width, is 300 square feet. Therefore, we can write the equation as: \( w * (w + 5) = 300 \)

Step 2: Solve the equation

To solve the equation for \( w \), start by rearranging the equation from step 1 to a standard quadratic form: \( w^2 + 5w - 300 = 0 \). Solving this quadratic equation using the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where a=1, b=5, c=-300, we have: \( w = \frac{-5 \pm \sqrt{5^2 - 4*1*(-300)}}{2*1} \). This yields two roots, \( w=15 \) or \( w=-20 \) but since width can't be negative, \( w=15 \) feet is the solution.

Step 3: Determine the length

To find the length of the garden, substitute \( w=15 \) feet into the length expression, yielding \( L= w + 5 = 15 + 5 = 20 \) feet.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the context of our rectangular garden problem, we define the width of the garden as a variable, denoted by w.

Let's illustrate the importance of choosing the right variable and setting up the expression accurately. The width w is a representation of any possible number that can be the width of the garden. By relating the length as w + 5, we create another algebraic expression that succinctly shows the relationship between length and width; it tells us the length is always 5 feet more than the width.

Understanding how to set up expressions like these is crucial, as it paves the way for forming an equation that can be solved to find the unknown variables. When dealing with rectangular area problems or any geometry problems, it's a good practice to denote your variables with clear, descriptive symbols that correspond to the dimensions or quantities you're trying to find.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ax^2 + bx + c = 0. It provides a direct method to find the roots of any quadratic equation. The formula is given by: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our garden problem, once we've established the algebraic expression reflecting the area of the rectangle and rearranged it into standard quadratic form, we apply the quadratic formula. With a=1, b=5, and c=-300, we substitute these values into the formula to find the possible values for w, the width.

The quadratic formula's versatility is shown in its ability to produce real and complex roots, depending on the discriminant (\(b^2 - 4ac\)). In our scenario, the discriminant is positive, which means we get two real solutions, one of which will be the physical width of our garden.

Solving Quadratic Equations

Solving quadratic equations is a critical skill in algebra. After formulating a quadratic equation from a real-world problem, as we did with the area of a garden, we move on to finding its solutions.

The quadratic formula is one route to this end, but it's not the only one. You might also factor the quadratic expression, complete the square, or even graph the equation to find the roots. In our garden's case, the quadratic formula was efficient.

Considering the physical context is essential after solving the equation. We obtained two roots, \(w=15\) and \(w=-20\), but we must dismiss \(w=-20\) as a garden can't have a negative width. This logical interpretation of mathematical solutions is key for successfully solving real-world problems with algebra.

Rectangular Area Problems

Rectangular area problems involve finding the length and width of a rectangle given its area or vice versa. They are practical applications of algebraic expressions and quadratic equations.

Our garden problem is a typical example where the relationship between length, width, and area must be understood. The area of a rectangle is found by multiplying its length by its width (\(Area = length \times width\)). These problems often lead to quadratic equations, as seen with the equation \(w(w+5) = 300\), since both dimensions are unknown and one is defined in terms of the other.

In solving these problems, once the width \(w\) is found, it's simple to compute the length by using the established relationship. Remember that context is important: the solutions must make sense in real-world terms, which means dismissing any negative dimensions, as they are not physically meaningful.