Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Geometry A room is \(1.5\) times as long as it is wide, and its perimeter is 75 feet (see figure). Find the dimensions of the room.

Step-by-Step Solution

Answer

The dimensions of the room are 15 feet (width) by 22.5 feet (length).

Step 1: Define Variables

Let \(x\) represent the width of the room. Then, the length of the room is \(1.5x\).

Step 2: Set up Equation using Perimeter

The formula for the perimeter of a rectangle is \(2(\text{length} + \text{width})\). Substitute the given perimeter and the expressions for length and width in this formula we get, \(2(x + 1.5x) = 75\).

Step 3: Solve the Equation

First simplify the equation, \(2(2.5x) = 75\), which gives \(5x = 75\). Now divide by 5 on both sides of the equation to solve for x, which gives \(x = 15\).

Step 4: Find the Length

Substitute \(x = 15\) into the expression for the length, gives \(1.5*15 = 22.5\).

Step 5: Returning to the problem's units

In the context of this problem, \(x = 15\) represents the width and \(1.5x = 22.5\) represents the length of the room. Both of these measures are in feet.

Key Concepts

Perimeter of a Rectangle

Understanding the perimeter of a rectangle is crucial in tackling many geometry problems. The perimeter is the total distance around the outside of a rectangle. This is calculated by adding up all the sides. In rectangles, opposite sides are equal in length. Hence, the formula for the perimeter is given by:

• \[P = 2( ext{length} + ext{width})\]

This formula uses the basic properties of a rectangle, where you double the sum of the length and width. It's like going around the rectangle once. Applying this, as we did with our problem, helps us relate the given perimeter, which was 75 feet, back to the dimensions, which are expressed in terms of variables.

Variable Substitution

Variables are used in mathematics to represent unknown values. Substitution is the process of replacing these variables with actual numbers or expressions.

In this problem, we defined a variable \(x\) to represent the width. Since we knew the length was 1.5 times the width, we substituted \(1.5x\) for the length in our equation for the perimeter. This makes use of the relationship between width and length, allowing us to express everything in terms of one variable.

• This simplification through substitution allows easy manipulation and calculation.
• Ensures that we use established relationships, like the 1.5 times rule, effectively.

Without substitution, solving the problem would be unnecessarily complicated. Substitution streamlines processes in mathematics significantly.

Equation Solving

Solving equations is a fundamental skill needed to uncover unknown values from algebraic expressions. Once we set up our equation with substitution, we need to simplify and solve it to find the value of the variable.
First, simplify by combining like terms or using basic arithmetic:

• From \(2(x + 1.5x) = 75\), we simplify to \(2(2.5x) = 75\).
• Our goal here is to isolate \(x\) on one side of the equation.

Finally, perform operations like division to solve:

• Divide both sides by 5: \(5x = 75\) becomes \(x = 15\).
• This tells us the width is 15 feet.

Solving equations involves carefully following methods that simplify the expression down to a concrete answer.

Algebraic Expressions

Algebraic expressions allow us to represent and solve for unknown values using known mathematical relationships. These expressions use variables, numbers, and operational symbols.

Our problem utilized the expression \(1.5x\) to represent the length of the room. This expression directly stems from the geometric relationship given:

• It simplifies calculations by providing one variable's relationship to another.
• It integrates into the perimeter equation smoothly for effective solution finding.

When working with algebraic expressions, understanding and manipulating them correctly is vital for effective problem-solving in geometry and beyond. By translating verbal descriptions into algebraic terms, we can solve complex real-world problems seamlessly.