Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

A rectangle has a width of 46 centimeters and a perimeter of 208 centimeters. What is the rectangle's length?

Step-by-Step Solution

Answer

The rectangle's length is 58 centimeters.

Step 1: Understand the Problem

We are given the width and the perimeter of a rectangle. To find its length, we use the formula for perimeter, which is 2*(length + width). The width is 46 centimeters, and the perimeter is 208 centimeters. We can write the equation as: 2*(L+46) = 208. Where L is the length we want to find.

Step 2: Solve for Length

First, divide both sides of the equation by 2 to simplify it. That gives us: L + 46 = 104. Then, to isolate L, we subtract 46 from both sides of the equation. That gives us: L = 104 - 46.

Step 3: Calculate Length

Finally, perform the subtraction: 104 - 46 = 58. Therefore, the rectangle's length is 58 centimeters.

Key Concepts

Understanding Perimeter

The perimeter of a shape is the total distance around its edges. For a rectangle, this involves adding up all sides of the four-sided figure. A rectangle has two pairs of equal sides, typically described as "length" and "width." To find the perimeter of a rectangle, we can use the formula:

• Perimeter = 2 × (Length + Width)

By multiplying the sum of the length and width by two, you account for both pairs of sides. This formula helps us determine how much border material would be needed if you were, for example, putting a fence around a rectangle-shaped yard. Knowing how to calculate perimeter is an essential skill in geometry, particularly when dealing with real-world applications such as construction and design projects.

Recognizing Properties of a Rectangle

A rectangle is a type of quadrilateral, which means it has four sides. The unique feature of a rectangle is that all four of its angles are right angles, or 90 degrees. An important characteristic of rectangles is that the opposite sides are equal in length.
When calculating dimensions such as perimeter or area, this property of equal opposite sides simplifies the process. In this particular exercise, knowing the width of the rectangle is essential because it allows us to use these properties to find the length when given the perimeter. By establishing one measurement, like the width, you can unlock the ability to find other measurements through basic arithmetic, reinforcing the strong relationship in geometry between known and unknown quantities.

Performing Length Calculations

Calculating the length of a rectangle when the width and perimeter are known involves a simple rearrangement of the standard perimeter formula. Here’s how you can solve for the length:

• Start with the formula for the perimeter: \[ 2 \times (L + W) = P \], where \( L \) is the length, \( W \) is the width, and \( P \) is the perimeter.
• Substitute the known values into the equation. For example, if \( W = 46 \) cm and \( P = 208 \) cm, then you write: \[ 2 \times (L + 46) = 208 \].
• Divide both sides by 2 to simplify: \[ L + 46 = 104 \].
• Finally, isolate \( L \) by subtracting the width from both sides: \[ L = 104 - 46 \].
• Calculate the result: \( L = 58 \) cm.

By understanding these steps, one can easily find the unknown dimension of a rectangle, using logical reasoning and basic math operations.