Problem 42 - Algebra and Trigonometry

Question

Dimensions of a Poster \(\quad\) A poster has a rectangular printed area 100 \(\mathrm{cm}\) by 140 \(\mathrm{cm}\) and a blank strip of uniform width around the edges. The perimeter of the poster is 1\(\frac{1}{2}\) times the perimeter of the printed area. What is the width of the blank strip?

Step-by-Step Solution

Answer

The width of the blank strip is 30 cm.

Step 1: Understand the Problem

The problem describes a poster with a rectangular printed area measuring 100 cm by 140 cm, surrounded by a blank strip of uniform width. The goal is to find the width of the blank strip given that the perimeter of the entire poster is 1.5 times the perimeter of the printed area.

Step 2: Calculate the Perimeter of the Printed Area

First, calculate the perimeter of the printed area. The formula for the perimeter of a rectangle is:\[ P = 2 \times (\text{length} + \text{width}) \]Substituting the given values for the printed area:\[ P = 2 \times (100 + 140) = 2 \times 240 = 480 \text{ cm} \]

Step 3: Express the Perimeter of the Entire Poster

Let the width of the blank strip be \(x\) cm. The dimensions of the entire poster, including the strip, would be \((100 + 2x)\) and \((140 + 2x)\). The perimeter of the entire poster is:\[ P' = 2 \times ((100 + 2x) + (140 + 2x)) = 2 \times (240 + 4x) = 480 + 8x \]

Step 4: Set Up the Perimeter Relationship Equation

According to the problem, the perimeter of the poster is 1.5 times the perimeter of the printed area. Therefore, set up the equation:\[ 480 + 8x = 1.5 \times 480 \]Simplify the right side:\[ 1.5 \times 480 = 720 \]Substitute back into the equation:\[ 480 + 8x = 720 \]

Step 5: Solve for the Width of the Strip

Rearrange the equation to solve for \(x\):\[ 8x = 720 - 480 \]\[ 8x = 240 \]\[ x = \frac{240}{8} = 30 \]

Step 6: Conclusion: Identify the Width of the Blank Strip

The calculations indicate that the width of the blank strip around the printed area is 30 cm.

Key Concepts

Perimeter

Understanding how the perimeter is calculated is crucial for solving problems in rectangular geometry. Perimeter is essentially the total length around a shape. For a rectangle, this is calculated using the formula:

• \( P = 2 \times (\text{length} + \text{width}) \)

In the given problem, the printed area of the poster, which measures 100 cm by 140 cm, helps us calculate its perimeter. First, we add the length and width:

• \( 100 + 140 = 240 \text{ cm} \)

Then, we multiply by 2 to get the total perimeter of the printed area:

• \( P = 2 \times 240 = 480 \text{ cm} \)

This calculated perimeter gives us a foundation for finding the relationship between the printed and full poster area. Remember that perimeter tells you the boundary measure but does not give information about the area inside that boundary.

Algebraic Equations

Algebraic equations form the backbone of problem-solving in mathematics. In the exercise you've seen, we deal with an equation that describes the relationship between the printed area and the entire poster. By introducing a variable \( x \) for the width of the blank strip, we can form expressions representing the whole dimensions of the poster:

• The length becomes \( 100 + 2x \)
• The width becomes \( 140 + 2x \)

With these, the perimeter of the entire poster is expressed as:

• \( P' = 2 \times ((100 + 2x) + (140 + 2x)) = 480 + 8x \)

The next step involves setting this equal to 1.5 times the calculated perimeter of the printed area, which involves solving the equation:

• \( 480 + 8x = 720 \)

This equation allows us to isolate \( x \) and find out its value. Algebraic manipulation like this one is a fundamental skill, enabling you to solve real-world and theoretical problems, achieving solutions by expressing relationships in mathematical terms.

Problem-Solving Steps

Effective problem-solving in geometry often involves breaking down exercises into manageable steps. This approach not only makes it easier to follow but also ensures you don't miss out on critical parts of the solution. Here’s a breakdown of how to tackle similar problems:

• Understand the problem: Identify all given information and what is being asked.
• Calculate known values: Start with what you can directly compute. Here, it was the perimeter of the printed area.
• Express unknowns: Use variables for unknown quantities and express all parts of the shape in terms of these variables.
• Formulate the equation: Set up relationships using the known formulas and given conditions in the problem.
• Solve the equation: Use algebraic methods to solve for the unknowns.
• Review your solution: Ensure all steps relate back to the problem and that the solution is reasonable.

These systematic steps guide you through a logical progression of thought, turning complex problems into a series of simpler tasks. This strategy helps in not only geometry but any mathematical problem, as it fosters a clear path from problem statement to solution.