Problem 255 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

Step-by-Step Solution

Answer

The width is 23 meters.

Step 1: Identify Given Information

The problem states that the area of the rectangle is 414 square meters and the length is 18 meters.

Step 2: Recall the Formula for Area of a Rectangle

The formula for the area of a rectangle is: \[ \text{Area} = \text{Length} \times \text{Width} \]

Step 3: Substitute the Given Values

Substitute the given values into the area formula: \[ 414 = 18 \times \text{Width} \]

Step 4: Solve for the Width

To find the width, divide both sides of the equation by 18: \[ \text{Width} = \frac{414}{18} \]

Step 5: Simplify the Calculation

Perform the division to solve for the width: \[ \text{Width} = 23 \]

Key Concepts

Area of a Rectangle

Understanding the area of a rectangle is essential in solving many geometry problems. The area represents the amount of space inside the rectangle, and it is measured in square units, such as square meters. The formula to calculate the area is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
In this formula:

• Length is the longer side of the rectangle.
• Width is the shorter side of the rectangle.

To solve for one of the dimensions when the area and the other dimension are known, you simply rearrange the formula. In our exercise, we know the area (414 square meters) and the length (18 meters), and we need to find the width. By substituting the known values into the formula, we can solve the equation to find the width.

Algebraic Equation

Algebraic equations often show up when solving geometry problems. An equation is a mathematical statement that asserts the equality of two expressions. In this exercise, by knowing the area and the length, we form an equation to find the width:
\[ 414 = 18 \times \text{Width} \]
This is a simple algebraic equation where you solve for the unknown width. Here's how to solve such an equation step-by-step:

• Isolate the variable (Width) by performing inverse operations.
• Divide both sides by 18: \[ \text{Width} = \frac{414}{18} \]
• Simplify the result to get the value for Width.

Understanding how to manipulate algebraic equations is a valuable skill, not only in geometry but across many areas of mathematics.

Geometry

Geometry involves studying shapes, sizes, and the properties of space. Solving problems like the one in our exercise helps you understand important geometric concepts. Rectangles, in particular, are foundational shapes in geometry.
Important attributes of rectangles include:

• Opposite sides are equal in length.
• Each angle is a right angle (90 degrees).

By applying these properties, combined with formulas for area and perimeter, we solve various problems and gain deeper insights into spatial relationships.
In our example, knowing the dimensions and properties of rectangles allows us to calculate unknown measures. As a result, we find the width by working through algebraic equations grounded in geometric principles.