Problem 12 - Beginning and Intermediate Algebra

Question

Write an equation and solve. A rectangular painting is twice as long as it is wide. Find its dimensions if it has an area of \(12.5 \mathrm{ft}^{2}\)

Step-by-Step Solution

Answer

The dimensions of the rectangular painting are \(2.5 \mathrm{ft}\) (width) by \(5 \mathrm{ft}\) (length).

Step 1: Let the width of the rectangle be x. Since the length is twice the width, its length will be 2x. #Step 2: Write the equation for area#

Given that the area of the rectangular painting is 12.5 square feet, we can write the equation for the area of the rectangle as: Area = length × width \(12.5 = 2x \times x\) #Step 3: Solve the equation#

Step 2: Simplify the equation: \(12.5 = 2x^2\) Now, divide both sides of the equation by 2: \(6.25 = x^2\) To find the value of x, take the square root of both sides of the equation: \(x = \sqrt{6.25}\) \(x = 2.5\) #Step 4: Find the dimensions of the rectangle#

The width of the rectangle is x, and the length is 2x. So: Width = \(2.5 \mathrm{ft}\) Length = 2 × \(2.5 \mathrm{ft}\) = \(5 \mathrm{ft}\) Hence, the dimensions of the rectangular painting are \(2.5 \mathrm{ft}\) (width) by \(5 \mathrm{ft}\) (length).

Key Concepts

Solving Equations

To solve an equation means we want to find the value of a variable that makes the equation true. In this problem, the variable represents the width of the rectangular painting. Often, you start with a known formula or condition. Then, you rearrange the formula, perform calculations, or apply mathematical rules to isolate the variable on one side of the equation.

The key steps to solve the equation for this problem are:

• Recognize the known quantities and the variable. Here, the area is known (12.5 square feet), and the width is unknown.
• Write the equation related to these quantities, which involves the geometric formula for the area of a rectangle.
• Simplify the equation to make it manageable and reduce it to solve for one variable.
• Once rearranged, manipulate the equation's sides (like dividing or taking square roots) to solve the equation.

Breaking down each step ensures you don't miss any crucial details when you solve the equation.

Geometric Dimensions

Geometric dimensions are measures that describe the size and shape of figures, such as rectangles. In our problem, these dimensions are the length and width of the rectangular painting. Understanding how these dimensions relate is crucial.

For rectangles:

• Width is often described as the shorter side.
• Length is twice the width, per the exercise condition. This relationship is critical for setting up your equation.
• Geometric dimensions relate directly to the formulas used to calculate properties like area or perimeter.

By identifying these dimensions and relationships upfront, we can confidently apply formulas and solve for unknown values using algebra.

Rectangular Area

The area of a rectangle is determined by its length and width, specifically through the formula, Area = Length × Width. This formula is a cornerstone of geometry and algebra when dealing with shapes.

In this problem, the area given is crucial to finding the unknown dimensions:

• The area is calculated by multiplying the known length and width of the rectangle.
• In algebraic problems, if you know the area and one dimension (or a relationship like the length being twice the width), you can set up an equation to find the missing dimension.
• The known area (12.5 square feet in this example) helps to find the dimensions through rearrangement and simplification of the equation.

Mastering how the area relates to the dimensions is essential for solving problems involving areas of rectangles and other geometric shapes.