Problem 25 - Fundamentals of Mathematics

Question

Find the length and width of a rectangle with perimeter 18 feet, if the width of the rectangle is 7 feet less than three times the length.

Step-by-Step Solution

Answer

The rectangle's length is 4 feet and width is 5 feet.

Step 1: Define the Variables

Let's define the variables for the problem. Let \( L \) represent the length of the rectangle, and \( W \) represent the width. According to the problem, the width of the rectangle is 7 feet less than three times the length, so \( W = 3L - 7 \).

Step 2: Write the Perimeter Equation

The perimeter of a rectangle is given by the formula \( P = 2L + 2W \). We know the perimeter is 18 feet. So, we have:\[2L + 2W = 18\]

Step 3: Substitute the Expression for Width

We already know that \( W = 3L - 7 \) from Step 1. Substitute this expression for \( W \) into the perimeter equation:\[2L + 2(3L - 7) = 18\]

Step 4: Simplify the Equation

Expand and simplify the equation:\[2L + 6L - 14 = 18\]Combine like terms:\[8L - 14 = 18\]

Step 5: Solve for Length

Add 14 to both sides of the equation:\[8L = 32\]Divide both sides by 8 to solve for \( L \):\[L = 4\]

Step 6: Solve for Width

Now that we know \( L = 4 \), substitute this value back into the expression for \( W = 3L - 7 \):\[W = 3(4) - 7 = 12 - 7 = 5\]

Step 7: Verify the Solution

Plug these values back into the perimeter formula to verify:\[P = 2(4) + 2(5) = 8 + 10 = 18\]Since the perimeter equals 18 feet, the solution is verified.

Key Concepts

Solving Linear Equations

Solving linear equations is a crucial skill in mathematics, especially in problem-solving scenarios involving unknown values. A linear equation is an equation where the highest power of the variable(s) involved is one, making it straightforward to solve. The process for solving a linear equation generally involves:

• Isolating the variable on one side of the equation: This often requires performing inverse operations - such as addition or subtraction - to both sides of the equation to keep it balanced.
• Simplifying both sides of the equation: Combine like terms and simplify expressions as needed.
• Solving for the variable: Once the equation is simplified, solve for the unknown variable by performing any necessary operations.

In our exercise, we defined the relationship between length and width of a rectangle through linear equations and used substitution to find the length and width, successfully applying these fundamental steps to arrive at the solution.

Geometry Problem-Solving

Geometry problem-solving often involves understanding the properties of shapes and how these can be expressed mathematically. For rectangles, key attributes include length, width, and perimeter.A rectangle's perimeter can be computed using the formula:\[ P = 2L + 2W \]Where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width. For this exercise, the given perimeter was 18 feet. By connecting this information with other given data, such as the width being 7 feet less than three times the length, you can derive necessary equations to solve the problem.

• Apply geometric formulas: Understand and utilize formulas to establish relationships between elements of the problem.
• Translate descriptions into mathematics: Use given information to create and solve mathematical equations.

This approach facilitates a step-by-step path to finding solutions to geometric problems by effectively translating real-world geometry scenarios into mathematical expressions.

Variable Substitution

Variable substitution is a technique used in algebra to simplify equations and make them easier to solve. This involves replacing a variable with an expression representing it, usually based on additional information provided in the problem.In our rectangle problem, the relationship between the length \( L \) and the width \( W \) is given as:\[ W = 3L - 7 \]We substituted \( W \) with this expression in the perimeter equation \( 2L + 2W = 18 \). This transformed the initial problem into a simpler equation that could be solved for one variable at a time.

• Ensure consistency: When substituting, ensure that all related terms are replaced correctly to maintain equation validity.
• Simplify and solve: Once substitution is performed, continue simplifying the equation to isolate the variable and find solutions.

Variable substitution is particularly useful for complex equations, enabling a clearer path to solving by reducing the number of unknowns and making equations more manageable.