Problem 24 - Algebra 2 and Trigonometry

Question

The length of a rectangle is 5 feet more than twice the width. a. If \(x\) represents the width of the rectangle, represent the perimeter of the rectangle in terms of \(x .\) b. If the perimeter of the rectangle is 2 feet more than eight times the width of the rec- tangle, find the dimensions of the rectangle.

Step-by-Step Solution

Answer

The width is 4 feet and the length is 13 feet.

Step 1: Understanding the Problem

We are given that the length of a rectangle is 5 feet more than twice the width. Let the width be represented by \(x\). Hence, the length can be written as \(2x + 5\). The perimeter \(P\) of a rectangle is given by the formula \(P = 2(l + w)\), where \(l\) is the length, and \(w\) is the width. We can now translate these into an expression and equation.

Step 2: Expressing Perimeter in Terms of Width

Using the width \(x\) and the given expression for the length \(2x + 5\), the perimeter \(P\) in terms of \(x\) can be written as: \[ P = 2((2x + 5) + x) \] \[ P = 2(3x + 5) \] \[ P = 6x + 10 \]

Step 3: Setting the Perimeter Equation

We are also told that the perimeter is 2 feet more than eight times the width. This gives us another equation: \[ P = 8x + 2 \] We have two expressions for \(P\): one from the perimeter formula, \(P = 6x + 10\), and our new equation \(P = 8x + 2\).

Step 4: Solving for Width \(x\)

Setting the two expressions for \(P\) equal to each other, we have: \[ 6x + 10 = 8x + 2 \] Rearrange this equation to solve for \(x\): \[ 10 - 2 = 8x - 6x \] \[ 8 = 2x \] \[ x = 4 \] Thus, the width \(x\) of the rectangle is 4 feet.

Step 5: Finding the Length Using Width

Now that we know \(x = 4\), substitute it back into the length expression: \[ l = 2x + 5 \] \[ l = 2(4) + 5 \] \[ l = 8 + 5 \] \[ l = 13 \] Hence, the length of the rectangle is 13 feet.

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the edges of the shape. It's like walking around the entire boundary. To calculate this, we add up all the side lengths. For a rectangle, this involves summing the length (\( l \)) and width (\( w \)), and then multiplying by 2 because a rectangle has two pairs of equal sides. The formula for the perimeter \( P \) is given by:\[ P = 2(l + w) \]Let's break it down:

• \( l \) is the length of the rectangle.
• \( w \) is the width.
• You add these together: \( l + w \).
• The result gives you one half of the perimeter since it includes one length and one width.
• Multiplying by 2 accounts for both pairs of sides.

Understanding this formula is crucial as it helps in deriving expressions for more complex problems involving rectangles.

Algebraic Expressions

Algebraic expressions allow us to use variables instead of fixed numbers, making calculations flexible. In the context of a rectangle, if a variable \( x \) represents the width, it means every mention of \( x \) will refer to the width. Given expressions like the length being "5 feet more than twice the width" can be translated into an equation.Here's how:

• If \( x \) is the width, "twice the width" is \( 2x \).
• Now, adding "5 feet more," we get \( 2x + 5 \) for the length.

This representation is important because it maintains accuracy through variable relationships, allowing substitution and solving different equations. Such expressions are fundamental in relating different dimensions of geometric figures mathematically.

Solving Equations

Solving equations is like solving a puzzle. It involves finding the unknown variable that makes the equation true. Let's say we have an equation derived from a perimeter formula and a condition given in a problem. Here's a quick play-by-play:

• You have one equation for the perimeter expressed using the width, \( P = 6x + 10 \).
• Another given condition sets \( P = 8x + 2 \).
• Equate these to solve for \( x \), i.e., \( 6x + 10 = 8x + 2 \).
• Rearrange terms to isolate \( x \).
• You get \( x = 4 \).

By solving, we find the width, \( x \), making further problem-solving stages easier to handle. It involves rearranging terms, adding, subtracting across equality, and isolating the variable.

Mathematical Problem Solving

Mathematical problem-solving involves structured thinking to break down complex problems. Let’s walk through solving a problem about a rectangle's dimensions using step-by-step logic. Start by:

• Understanding the problem: Translate words into mathematical expressions. Here, relate dimensions with variables.
• Express the conditions: Represent the perimeter and length expressions using the width for better clarity.
• Set up appropriate equations: Relate these expressions as given by conditions, like perimeter equations.
• Solve: Use algebraic manipulation to find variable values like the width.
• Verify and interpret results: Check back with original conditions. If width is 4 feet and length is 13 feet, this should satisfy initial conditions.

Mathematical problem-solving requires logical thinking throughout each step and ensures each part connects seamlessly to the next, ultimately leading to a valid solution.