Problem 21 - Elementary Algebra

Question

The width of a rectangle is 1 foot longer than one half the length. Find the dimensions (lengh and width) of the rectangle if the perimeter is 44 feet.

Step-by-Step Solution

Answer

Answer: The length of the rectangle is 14 feet, and the width of the rectangle is 8 feet.

Step 1: Write down the given information

We are given that the width of a rectangle is 1 foot longer than one half the length, and the perimeter is 44 feet. Let the length be represented by L and the width be represented by W. We can express the relationship between the width and length as: W = 0.5L + 1 We are also given that the perimeter (P) is 44 feet: P = 2(L+W) 44 = 2(L+W)

Step 2: Substitute W in the perimeter equation

We will substitute the expression for W (from the width-length relationship) into the perimeter equation: 44 = 2(L+(0.5L + 1))

Step 3: Solve for L

Now we need to solve the equation for L: 44 = 2(L + 0.5L + 1) 44 = 2(1.5L + 1) 22 = 1.5L + 1 Subtract 1 from both sides of the equation: 21 = 1.5L To solve for L, divide both sides by 1.5: L = 14

Step 4: Find W

Now that we have found the length (L), we can find the width (W) using the width-length relationship equation: W = 0.5L + 1 W = 0.5(14) + 1 W = 7 + 1 W = 8

Step 5: State the dimensions of the rectangle

The dimensions of the rectangle are: - Length (L) = 14 feet - Width (W) = 8 feet

Key Concepts

Algebraic Equations

Algebraic equations are powerful tools in problem-solving because they allow us to express relationships between different quantities. In this exercise, we use algebraic equations to find the dimensions of a rectangle. The primary equation given is the relationship between the width and length: \( W = 0.5L + 1 \). This formula shows that the width is one foot longer than half the length. By expressing one variable in terms of another, we can simplify complex relationships into manageable equations. When working with algebraic equations, a key strategy is to isolate the variable of interest, which is often achieved through substitution. For instance, by substituting the expression for \( W \) from the width-length relationship into the perimeter equation, we simplify the problem into a single equation with one variable: \( 44 = 2(L + (0.5L + 1)) \). Once the equation is simplified, we can easily solve for \( L \), and subsequently find \( W \) using the relationship \( W = 0.5L + 1 \). This systematic approach is common in algebraic problem-solving, demonstrating a clear path from complex interrelations to simple solutions.

Geometry Problems

Geometry problems, like finding the dimensions of a rectangle, often require visualizing shapes and understanding their properties. In this case, the rectangle's perimeter is given as 44 feet, which is the sum of all its sides twice: \( P = 2(L + W) \). Visualizing a rectangle with given numeric relationships helps to formulate equations, as it involves both length and width. The challenge here is to translate verbal descriptions, such as "the width is 1 foot longer than one half the length," into algebraic terms. This step is crucial in geometry problems because it crafts a bridge between geometric figures and algebraic symbols. Understanding how to manipulate and set up equations from geometric concepts is vital in solving geometry problems effectively. Knowing properties like area, perimeter, and the relationships between dimensions is the cornerstone in tackling similar exercises.

Problem Solving Steps

Breaking down a problem into clear steps is essential for finding effective solutions. Let's walk through the steps that were used in this exercise.

• **Step 1:** Begin by writing down all the given information and relationships. Here, the width was defined in terms of the length, and the perimeter was known.
• **Step 2:** Substitute known variables into the main formula involved. The perimeter equation was used to substitute the expression for width.
• **Step 3:** Solve for the unknown. The equation was simplified and solved for length by isolating \( L \).
• **Step 4:** Use the found value to calculate any other unknowns. Once \( L \) was known, \( W \) was easily calculated using the initial equation.
• **Step 5:** Clearly state the solution, verifying it with the original problem requirements.

These structured steps not only guide towards the solution but also cultivate a clear problem-solving mindset, essential for tackling any math problem effectively.