Problem 33 - Algebra and Trigonometry

Question

Framing a Painting \(\quad\) Al paints with watercolors on a sheet of paper 20 in. wide by 15 in. high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture?

Step-by-Step Solution

Answer

The mat strip is 4 inches wide.

Step 1: Determine the Length and Width of the Mat

The mat has a perimeter of 102 inches. Let the width of the strip of the mat be represented as \( x \). Then, the total width of the mat can be expressed as \( 20 + 2x \), and the total height can be expressed as \( 15 + 2x \). The perimeter formula is given by \( 2 \times ( ext{length} + ext{width} ) \).

Step 2: Set Up the Equation for the Perimeter

Using the perimeter formula, the equation for the perimeter of the mat can be set up as:\[2((20 + 2x) + (15 + 2x)) = 102\]

Step 3: Simplify and Solve the Equation

First, simplify the expression within the parentheses:\[(20 + 2x) + (15 + 2x) = 35 + 4x\]Now, substitute back into the perimeter equation:\[2(35 + 4x) = 102\]Simplify further:\[70 + 8x = 102\]Subtract 70 from both sides:\[8x = 32\]Finally, divide both sides by 8:\[x = 4\]

Step 4: Interpret the Solution

The value of \( x \) represents the width of the strip of the mat showing around the picture. Therefore, the width of the strip is 4 inches.

Key Concepts

Perimeter

When we talk about the perimeter, we mean the total distance around the edge of a two-dimensional shape. It's like wrapping a string around the shape's outer edges and then measuring the length of that string. In this exercise, we are dealing with the perimeter of a mat that has a painting placed on it.

For the mat, you can think of it as having a rectangular shape where the perimeter can be calculated using the formula:

• The formula for perimeter: \[ P = 2 imes ( ext{length} + ext{width} ) \]

The given problem provides the perimeter as 102 inches. We use this information to find the width of the strip surrounding Al's painting. Understanding perimeter helps you visualize how changes in dimensions (length or width) affect the total boundary length.

Mathematical equation

A mathematical equation is like a scale that balances two expressions. Equations show relationships between numbers and variables, which in our exercise, helps us determine unknown values.

In the painting problem, we set up an equation to represent the perimeter of the mat. The equation provided was:

• \[ 2 imes ((20 + 2x) + (15 + 2x)) = 102 \]

Here, the expressions inside parentheses \((20 + 2x)\) and \((15 + 2x)\) represent the total dimensions of the mat, inclusive of the strips around the painting. The equation signifies that when you double the sum of the length and width, you get the full perimeter. It underscores how equations model real-world situations mathematically.

Linear equation

Linear equations are a special type of equation where the variable (in this case, \( x \)) is raised to the power of one, meaning there are no squares, cubes, or higher powers of \( x \).

A linear equation models a straight line when plotted on a graph. They're particularly popular because they are easy to solve and understand. In our strip problem, the equation simplifies into a linear form:

• \[ 70 + 8x = 102 \]

To solve linear equations, we typically simplify them step by step until the variable is isolated. This is done by first subtracting or adding terms on both sides of the equation, followed by multiplying or dividing as needed to solve for \( x \). For linear equations, solving them can visually be similar to "undoing" the operations that have been applied to \( x \).

Problem solving

Problem-solving is a vital skill in mathematics, involving a methodical approach to tackle and fix challenges. This skill requires breaking down the problem into smaller, manageable parts and solving each one step by step. Let's explore its application in our scenario:

• Identify what is known and what needs to be found. Here, we knew the perimeter, but needed the strip's width.
• Translate the problem into a mathematical equation or equations.
• Simplify and solve these equations to find the unknowns.
• Finally, interpret what the solution means in the context of the problem.

By understanding the components and using clear logical reasoning, problem-solving becomes easier. Plus, familiarizing yourself with various types of problems and practicing more enhances your capability to solve them effectively.