Problem 26 - Elementary Algebra

Question

The length of a rectangle is 2 feet more than its width. If the area of the rectangle is 48 square feet, then find the length and width.

Step-by-Step Solution

Answer

The width is 6 feet and the length is 8 feet.

Step 1: Define variables

Let the width of the rectangle be \( w \) feet. According to the problem, the length \( l \) is 2 feet more than the width. So, \( l = w + 2 \).

Step 2: Write the area formula

The area \( A \) of a rectangle is given by the formula \( A = l \times w \). We know the area is 48 square feet, so we can write the equation \( (w + 2) \times w = 48 \).

Step 3: Expand and rearrange the equation

Expand the equation from Step 2: \( w^2 + 2w = 48 \). Rearrange it into the standard quadratic form: \( w^2 + 2w - 48 = 0 \).

Step 4: Solve the quadratic equation

Use the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find \( w \), where \( a = 1 \), \( b = 2 \), and \( c = -48 \). The discriminant \( b^2 - 4ac = 2^2 - 4(1)(-48) = 4 + 192 = 196 \). Thus, \( w = \frac{-2 \pm \sqrt{196}}{2} \).

Step 5: Calculate the width

The square root of 196 is 14, so \( w = \frac{-2 + 14}{2} \) or \( w = \frac{-2 - 14}{2} \). Calculating these gives \( w = 6 \) (since width cannot be negative, we discard \( w = -8 \)).

Step 6: Find the length

Substitute \( w = 6 \) back into the expression for length, \( l = w + 2 \). Therefore, \( l = 6 + 2 = 8 \) feet.

Key Concepts

Rectangle Area

The area of a rectangle is a fundamental concept in geometry that helps us understand how much space is enclosed within the rectangle's boundaries. Imagine having a piece of land or a flat surface you'd like to cover with tiles.

The rectangle's area tells us how many tiles we'd need to completely cover it, if each tile is a square foot. The formula to determine this is fairly straightforward:

• Let length (\( l \)) be one side of the rectangle.
• Let width (\( w \)) be the other side.
• The product of these two dimensions gives the area: \[ A = l \times w \].

In the exercise, we're told that the area is 48 square feet. By understanding this equation, we can work backward to find other dimensions if needed. This involves the use of some algebraic manipulations.

Length and Width Calculation

Finding the length and width when given a rectangle's area requires understanding their relationship and solving equations. In this exercise, you start by assigning a variable to one of the dimensions.

Let's assume the width is (\( w \)), then according to the problem, the length is 2 feet more, which gives us (\( l = w + 2 \)).
This captures the relationship between the two dimensions. To find exact values, they are substituted into the area formula as follows:

• Use the equation \((w + 2) \times w = 48\).
• Expand to get \(w^2 + 2w = 48\).

This is a quadratic equation, a critical step toward finding the exact measurements. Once we rearrange it into the standard form \( w^2 + 2w - 48 = 0 \), the quadratic formula can be employed.

Area Formula

The area formula for a rectangle is pivotal in solving problems involving dimensions when given one of the quantities like length, width, or area. The formula \( A = l \times w \) is essential because it provides a direct correlation between dimensions and the area.

• The length \( l \) and width \( w \) are multiplied to result in the total square footage.
• In the given problem, we used this relationship to set up an equation to find unknown dimensions based on the area.

By transforming this formula into a quadratic equation, i.e., \( w^2 + 2w - 48 = 0 \), and solving it using the quadratic formula, dimensions that satisfy the area condition can be obtained. Understanding and practicing this formula solidifies one's ability to solve a variety of geometric problems.