Problem 272 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of a rectangle is five inches more than twice the width. The perimeter is 34 inches. Find the length and width.

Step-by-Step Solution

Answer

Width = 4 inches; Length = 13 inches.

Step 1: Define the variables

Let the width of the rectangle be denoted by \(x\text{ (inches)}\). Then the length of the rectangle would be \(5\) inches more than twice the width, i.e., \(2x + 5\).

Step 2: Perimeter formula

The perimeter of a rectangle can be found using the formula: \(P = 2 \times (\text{length} + \text{width})\). Given that the perimeter is \(34\) inches, we can write: \(34 = 2 \times ((2x + 5) + x)\).

Step 3: Simplify the equation

Simplify the equation as follows: \(34 = 2 \times (3x + 5)\). Thus, \(34 = 6x + 10\).

Step 4: Solve for x

Isolate \(x\) to solve for width of the rectangle.Subtract \(10\) from both sides: \(34 - 10 = 6x\). This simplifies to \(24 = 6x\). Divide both sides by \(6\): \(x = 4\).

Step 5: Calculate the length

Now, substitute \(x = 4\) back into the expression for length: \((2x + 5)\). Therefore, the length is \(2(4) + 5 = 8 + 5 = 13\).

Step 6: Verify the solution

To verify, substitute the calculated length and width back into the perimeter formula:\(P = 2 \times (13 + 4) = 2 \times 17 = 34\),which matches the given perimeter.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this rectangle perimeter problem, algebra helps us translate real-world scenarios into mathematical expressions. For example, when the problem states that the length of the rectangle is five inches more than twice the width, we express this relationship algebraically as: \( \text{length} = 2x + 5 \). Here, \( x \) represents the width of the rectangle. Algebraic equations like these allow us to find unknown values by manipulating the equations according to algebraic rules.
Understanding this conversion from words to equations is crucial. It helps us form the right mathematical relationships needed to solve problems effectively.

Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. In this problem, we are dealing with the rectangle, which is a basic geometric shape. A rectangle has four sides with opposite sides being equal in length. The concept of perimeter in geometry is particularly important here. The formula for the perimeter of a rectangle is: \[ P = 2(\text{length} + \text{width}) \]
Given that the total perimeter is 34 inches, we set up the equation and use geometry to understand the physical context of the problem. This is essential for visualizing what the problem is about and ensuring our algebraic manipulations correspond to real-world measures.

Solving Equations

Solving equations is a fundamental part of both algebra and geometry. It involves finding the value of the unknowns that make the equation true. In this rectangle perimeter problem, we have the equation: \[ 34 = 2((2x + 5) + x) \]
This equation stems from the perimeter formula. Solving it step-by-step includes:

• Combining like terms: Write it as \( 34 = 2(3x + 5) \).
• Expanding and simplifying: Distribute the 2 to get \(34 = 6x + 10 \).
• Isolating the variable: Subtract 10 from both sides to get \(24 = 6x \).
• Finding the unknown: Dividing by 6 to get \( x = 4 \), the width.

Substituting back to find the length gives us \( \text{length} = 2(4) + 5 = 13 \) inches. Verifying our answer by plugging these values back into the perimeter equation confirms they are correct. This methodological approach is vital for tackling similar problems and is central to mathematical problem-solving.