Problem 46 - College Algebra

Question

The length of a rectangular pool is 6 meters less than twice the width. If the pool's perimeter is 126 meters, what are its dimensions? (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The dimensions of the pool are length \(L = 50\) meters and width \(W = 28\) meters.

Step 1: Formulate Relationship Between Length and Width

The problem states that the length \(L\) of the pool is 6 meters less than twice the width \(W\). This can be written as an equation: \(L = 2W - 6\).

Step 2: Formulate Perimeter Equation

The perimeter \(P\) of a rectangle is given by the formula \(P = 2L + 2W\). According to the problem, the perimeter of the pool is 126 meters. Substituting this value into the equation gives: \(126 = 2L + 2W\).

Step 3: Substitute the Relationship into the Perimeter Equation

Substitute the value of \(L\) from Step 1 into the perimeter equation to get: \(126 = 2(2W - 6) + 2W\).

Step 4: Solve for Width

Simplify and solve the equation from Step 3 to get the width \(W\) of the pool: \( W = 28\)meters.

Step 5: Find Length

Substitute the width \(W\) from Step 4 into the equation from Step 1 to find the length \(L\) of the pool: \(L = 2(28) - 6\)meters. So \(L = 50\)meters.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent numbers in equations. They allow us to find unknown quantities by expressing relationships between them. In the problem about the rectangular pool, we utilize a basic algebraic equation to find the relationship between the pool's length and width. To solve it:

• Identify what is known and unknown.
• Translate the word problem into an algebraic expression.

For the pool, the length is 6 meters less than twice the width. Hence, if we represent the width as \(W\), the length \(L\) can be expressed by the equation \(L = 2W - 6\). This equation encapsulates the relationship between the pool's dimensions, setting the stage for further calculations.

Perimeter of a Rectangle

The perimeter of a rectangle is the sum of all its sides. To calculate it, you simply add together the lengths and widths of the rectangle, then multiply by two, as the formula is \(P = 2L + 2W\).
The question gives a perimeter of 126 meters for the pool, which means:

• We use the perimeter formula to create another equation \(126 = 2L + 2W\).
• This formula and our previously found expression for \(L\) are used to form a system of equations.

By substituting the expression for \(L\) from the first algebraic equation into the perimeter equation, we can further solve for the width \(W\) of the pool.

Problem-Solving in Algebra

Problem-solving in algebra involves finding unknowns systematically using logical reasoning and arithmetic operations. With the pool problem, a structured approach helps to solve for the unknown dimensions efficiently. Here are some steps for effective problem-solving:

• Translate the word problem into mathematical equations using known formulas or definitions.
• Substitute known values to simplify the equations.
• Use algebraic manipulation to isolate the unknown variable.

For the pool, inserting the expression for \(L\) into \(126 = 2L + 2W\) results in \(126 = 2(2W - 6) + 2W\). Solving this simplifies the problem to find \(W = 28\). Once the width is known, substituting back yields the length.

Width and Length Relationship

Understanding the relationship between width and length is crucial for solving rectangular problems. In this case, the problem states a direct relationship between these dimensions: length is determined based on the width. Let's break it down:

• The given relationship is \(L = 2W - 6\), meaning to determine the length, you need the width.
• Once the width \(W\) is known to be 28 meters after solving the equations, plug it back into the expression to find \(L\).
• The length is calculated: \(L = 2(28) - 6 = 50\) meters.

Recognizing these relationships not only solves the perimeter problem but also builds intuition about how changes in one dimension affect the other.