Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

A rectangle has a width of 44 centimeters and a perimeter of 188 centimeters. What is the rectangle's length?

Step-by-Step Solution

Answer

The length of the rectangle is 50 centimeters.

Step 1: Analyze the Formula for Perimeter of a Rectangle

The formula for the perimeter of a rectangle is \(2 \times (length + width)\). Here, 'length' and 'width' are the lengths of the sides of the rectangle, and the number '2' represents the two pairs of opposite sides.

Step 2: Substituting the Given Values into the Formula

We substitute the given values into the formula. We have the perimeter (P) as 188 cm and the width (W) as 44 cm. So the formula becomes: \(188 = 2 \times (L + 44)\). 'L' is the length that we need to find.

Step 3: Simplify Equation

Divide both sides of the equation by 2. This gives \(L + 44 = 94\).

Step 4: Solving for the Length

Finally, solving for 'L' we get \(L = 94 - 44 = 50\). So the length of the rectangle is 50 cm.

Key Concepts

Understanding the Basics of Geometry

Geometry is a fascinating branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. Rectangles are a fundamental shape in geometry. By studying rectangles, you can learn a lot about geometry basics. Rectangles are everywhere, found in everyday objects like books and screens.
In geometry, shapes like rectangles are defined by their sides and angles. Rectangles have four sides and four right angles. This ensures all angles in a rectangle are 90 degrees. They are part of a larger group called quadrilaterals, which are polygons with four sides. In a rectangle, opposite sides are not only parallel but also equal in length. This is a key property that distinguishes rectangles from other quadrilaterals.
By studying geometry, you develop spatial understanding and problem-solving skills. It's not only about calculations but also about understanding the spatial arrangements and structures around us.

Calculating Length in Rectangles

When you're asked to find the length of a rectangle, you often use known values like width or perimeter. The length is the longer side of the rectangle, while the width is the shorter one. The formula for the perimeter \[P = 2 \times (L + W)\]helps us find the length when the perimeter and width are given.
Here's a simple process to calculate the length:

• Start with the perimeter formula: \(2 \times (L + W)\).
• Substitute the known perimeter and width into the formula.
• Simplify the equation by dividing by 2 to isolate the term \(L + W\).
• Solve the resulting linear equation to find the unknown length \(L\).

Using this step-by-step approach ensures you systematically solve for the length, avoiding mistakes and confirming the rectangle's measurements are accurate.

Exploring Rectangle Properties

The properties of rectangles are what make them distinct and recognizable. Knowing these properties helps in solving geometric problems effectively. Let's look at some unique features of rectangles:

• Opposite Sides are Equal: In a rectangle, the opposite sides are always equal in length. This is crucial for calculating the perimeter and area.
• Right Angles: Every angle in a rectangle is 90 degrees. This consistent angle measurement ensures the shape is a rectangle.
• Diagonals are Equal: The diagonals of a rectangle are equal in length, and they bisect each other. This can be helpful for advanced geometry problems.

These properties come in handy when dealing with rectangle calculations. They simplify the process of calculating dimensions, confirming the shape, and solving complex problems. Understanding these core properties of rectangles is essential in geometry and can be applied in real-world contexts.