Question

Dimensions of a Window The area of a rectangular window is to be 306 square centimeters. If the length exceeds the width by 1 centimeter, what are the dimensions?

Short answer

The width is 17 cm and the length is 18 cm.

Step-by-step solution

Identify the Variables

Let the width of the window be denoted by \(w\) centimeters. Consequently, the length will be \(w + 1\) centimeters since it exceeds the width by 1 centimeter.

Set Up the Equation

The area of the rectangle is given as 306 square centimeters. The area \(A\) of a rectangle is given by the formula \(A = \text{length} \times \text{width}\). Substituting the known values, we get: \[ \(w + 1\) \times w = 306 \]

Form the Quadratic Equation

Simplify the equation \[ w(w + 1) = 306 \] to form a quadratic equation: \[ w^2 + w = 306 \]

Solve the Quadratic Equation

Re-arrange the quadratic equation into standard form: \[ w^2 + w - 306 = 0 \]. Solve this using the quadratic formula \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where \(a = 1\), \(b = 1\), and \(c = -306\).

Calculate the Discriminant

The discriminant \( \Delta = b^2 - 4ac \): \[ \Delta = 1^2 - 4(1)(-306) = 1 + 1224 = 1225 \]

Find the Roots

Substitute the discriminant into the quadratic formula: \[ w = \frac{-1 \pm \sqrt{1225}}{2} = \frac{-1 \pm 35}{2} \]. This gives two solutions: \[ w = \frac{34}{2} = 17 \] and \[ w = \frac{-36}{2} = -18 \]. Since width cannot be negative, we discard \( w = -18 \). Thus, \( w = 17 \).

Find the Length

The length is \( w + 1 = 17 + 1 = 18 \) centimeters.

Verify the Solution

Check if the calculated dimensions satisfy the given area: \( 17 \times 18 = 306 \). The calculations are correct.

Key concepts

solving quadratic equations

Quadratic equations are fundamental in algebra and appear in various problems, including those involving physical shapes and areas. A quadratic equation typically has the form \[ ax^2 + bx + c = 0 \].
Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable we aim to solve for. The standard method to solve quadratic equations is using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
Another approach involves factoring the quadratic expression or completing the square. Regardless of the method, solving these equations helps find the values for \(x\) that satisfy the equation.

In practical problems like finding dimensions of a rectangular window, quadratic equations naturally arise. By understanding how to manipulate and solve these equations, you can determine important measurements efficiently.

rectangular area

The area of a rectangle is a fundamental geometric concept calculated as \( \text{length} \times \text{width} \).
In many real-world problems, the area may be known, requiring you to find the respective dimensions. In our example, the problem states that the area of the rectangular window is 306 square centimeters, with the length exceeding the width by 1 centimeter.

Using the given information, we let the width be \(w\) and the length be \(w + 1\). Plugging into the area formula \[ A = \text{length} \times \text{width} \text{= 306} \] yields the equation
\[ (w + 1) \times w = 306 \].
Simplifying further gives us the quadratic equation \( w^2 + w - 306 = 0 \), which we can solve to find the window's dimensions.

discriminant

The discriminant is a key component in solving quadratic equations, found under the square root in the quadratic formula: \( \text{b^2 - 4ac} \).
This value indicates the nature and number of roots of the equation. Specifically:

• If \( \text{b^2 - 4ac} > 0 \), we have two distinct real roots.
• If \( \text{b^2 - 4ac} = 0 \), there's exactly one real root (a repeated root).
• If \( \text{b^2 - 4ac} < 0 \), there are no real roots, only complex ones.

For our window dimensions example, substituting \(a = 1\), \(b = 1\), and \(c = -306\) into \( \text{b^2 - 4ac} \) gives \[ \text{1}^2 - 4(1)(-306) = 1 + 1224 = 1225 \].
This positive discriminant confirms two distinct real roots.
Subsequently, solving the equation using these roots helps determine the width and, consequently, the length of the window.