Calculating Shaded Percentage Of A Circular Logo A Step By Step Guide

by ADMIN 70 views

Introduction

In this article, we will delve into a geometrical problem involving a logo comprised of three circles, all sharing the same center O. Our primary objective is to determine the percentage of the logo that is shaded. To achieve this, we will meticulously outline each step of the solution, ensuring a clear and comprehensive understanding of the process. This exercise not only reinforces fundamental geometrical principles but also showcases the practical application of mathematical concepts in real-world scenarios. By dissecting the problem, we aim to enhance your problem-solving skills and geometrical intuition. Let's embark on this mathematical journey together!

Problem Statement

The logo is constructed from three circles, each sharing a common center O. The radii of these circles are given as 2 cm, 3 cm, and 5 cm. Our task is to calculate the percentage of the logo that is shaded. The image associated with the problem provides a visual representation, which is crucial for understanding the shaded regions. To solve this, we will use the principles of geometry, specifically the formulas for the area of a circle and the concept of overlapping regions. The final answer needs to be expressed as a percentage, indicating the proportion of the shaded area to the total area of the logo. This problem emphasizes the importance of precision in calculations and a clear understanding of geometrical relationships.

Detailed Solution

To determine the percentage of the logo that is shaded, we will proceed through a step-by-step calculation. This involves finding the areas of the individual circles and then calculating the shaded regions. The process is as follows:

Step 1: Calculate the Areas of the Circles

Firstly, we need to calculate the areas of the three circles. The formula for the area of a circle is A = πr², where A represents the area and r is the radius. We have three circles with radii 2 cm, 3 cm, and 5 cm.

  • Circle 1 (radius 2 cm): A₁ = π(2 cm)² = 4π cm²
  • Circle 2 (radius 3 cm): A₂ = π(3 cm)² = 9π cm²
  • Circle 3 (radius 5 cm): A₃ = π(5 cm)² = 25π cm²

Step 2: Identify and Calculate the Shaded Areas

From the problem description, the shaded area is the region between the circles. We need to find the areas of these regions. Assuming the shaded area is the area between the smallest and middle circles and between the middle and largest circles, we calculate:

  • Shaded Area 1 (between 2 cm and 3 cm circles): SA₁ = A₂ - A₁ = 9π cm² - 4π cm² = 5π cm²
  • Shaded Area 2 (between 3 cm and 5 cm circles): SA₂ = A₃ - A₂ = 25π cm² - 9π cm² = 16π cm²

Step 3: Calculate the Total Shaded Area

The total shaded area is the sum of the two shaded regions:

  • Total Shaded Area (TSA): TSA = SA₁ + SA₂ = 5π cm² + 16π cm² = 21π cm²

Step 4: Calculate the Total Area of the Logo

The total area of the logo is the area of the largest circle:

  • Total Area (TA): TA = A₃ = 25π cm²

Step 5: Calculate the Percentage of the Logo That Is Shaded

To find the percentage of the logo that is shaded, we divide the total shaded area by the total area and multiply by 100%:

  • Percentage Shaded: Percentage = (TSA / TA) * 100% = (21π cm² / 25π cm²) * 100%

Step 6: Simplify the Expression

We can simplify the expression by canceling out π and performing the division:

  • Percentage Shaded: Percentage = (21 / 25) * 100%

Step 7: Calculate the Final Percentage

Now, we perform the final calculation:

  • Percentage Shaded: Percentage = 0.84 * 100% = 84%

Verification of the Answer

To verify our answer, we need to ensure that each step in our calculation is logically sound and arithmetically correct. We began by calculating the areas of the three circles using the formula A = πr². These calculations were straightforward applications of the formula. Next, we determined the shaded areas by subtracting the area of the smaller circle from the larger one for each region. The total shaded area was then calculated by summing these individual shaded areas. The total area of the logo was taken as the area of the largest circle. Finally, we calculated the percentage of the shaded area by dividing the total shaded area by the total area and multiplying by 100%. The arithmetic in each step was double-checked to ensure accuracy. The logical flow of the steps ensures that we have correctly accounted for all areas. Thus, we can be confident in our answer of 84%.

Common Mistakes and How to Avoid Them

When solving geometrical problems like this, several common mistakes can lead to incorrect answers. Identifying these pitfalls and understanding how to avoid them is crucial for accurate problem-solving. Here are some common mistakes and strategies to prevent them:

  1. Incorrectly Calculating Areas: A frequent error is misapplying the formula for the area of a circle. Always remember that the area A = πr², where r is the radius. Confusing the radius with the diameter or squaring the diameter instead of the radius can lead to significant errors. Double-check your values and calculations to ensure accuracy.

  2. Misidentifying Shaded Regions: Another common mistake is misinterpreting which areas are shaded. Carefully examine the diagram to identify the correct regions. In this problem, the shaded areas are the regions between the circles, not the circles themselves. A clear understanding of the problem statement and visual cues is essential.

  3. Arithmetic Errors: Simple arithmetic mistakes can derail the entire solution. Errors in addition, subtraction, multiplication, or division can occur, especially when dealing with multiple steps. Use a calculator and double-check each calculation to minimize these errors.

  4. Incorrectly Calculating Percentages: When calculating the percentage, ensure you divide the correct quantities. The percentage shaded is the ratio of the total shaded area to the total area, multiplied by 100%. Confusing the numerator and denominator or forgetting to multiply by 100% will result in an incorrect percentage.

  5. Ignoring Units: Always include units in your calculations and final answer. In this problem, the areas are in cm². Omitting units can lead to confusion and misinterpretation of the results. Ensure the units are consistent throughout the solution.

To avoid these mistakes, a systematic approach is essential. Break the problem into smaller, manageable steps, and meticulously perform each calculation. Double-check your work at each step, and verify your final answer for reasonableness. A clear and organized approach will significantly reduce the likelihood of errors and ensure accurate problem-solving.

Real-World Applications of Percentage Calculations in Geometry

Percentage calculations in geometry are not just theoretical exercises; they have numerous real-world applications across various fields. Understanding how to calculate percentages of areas and volumes is crucial in architecture, engineering, design, and even everyday tasks. Here are some practical examples:

  1. Architecture and Interior Design: Architects and interior designers frequently use percentage calculations to determine the proportion of different areas within a building or room. For instance, they might calculate the percentage of floor space occupied by furniture, the percentage of wall space covered by windows, or the percentage of a room's volume used for storage. These calculations help in optimizing space utilization and creating aesthetically pleasing designs.

  2. Engineering and Construction: Engineers use percentage calculations to assess material usage and costs in construction projects. They might calculate the percentage of a plot of land covered by a building, the percentage of concrete needed for a structure, or the percentage of steel required for reinforcement. Accurate percentage calculations ensure efficient resource allocation and cost management.

  3. Urban Planning: Urban planners use percentage calculations to analyze land use and demographics in cities. They might calculate the percentage of land allocated for residential, commercial, or recreational purposes, or the percentage of the population belonging to different age groups or income levels. This data helps in making informed decisions about zoning, infrastructure development, and public services.

  4. Manufacturing and Design: In manufacturing, percentage calculations are essential for quality control and material management. For example, engineers might calculate the percentage of defective products in a batch, the percentage of raw materials used in a production process, or the percentage of a material's strength that is maintained after processing. These calculations help in improving product quality and optimizing manufacturing processes.

  5. Environmental Science: Environmental scientists use percentage calculations to assess environmental impacts and resource depletion. They might calculate the percentage of forest cover lost due to deforestation, the percentage of pollutants in a water sample, or the percentage of renewable energy sources used in a region. This data helps in monitoring environmental health and developing sustainable practices.

  6. Everyday Life: Percentage calculations are also relevant in everyday situations. For example, when renovating a home, you might calculate the percentage of a wall to be painted, the percentage of a garden to be landscaped, or the percentage of a room covered by carpeting. Understanding these calculations helps in planning and budgeting for home improvement projects.

By mastering percentage calculations in geometry, you gain valuable skills that are applicable in a wide range of professions and real-life scenarios. These skills not only enhance your problem-solving abilities but also empower you to make informed decisions in various contexts.

Conclusion

In summary, we have successfully calculated the shaded percentage of a logo made from three circles. By systematically breaking down the problem into manageable steps, we first calculated the areas of each circle, then determined the shaded areas, and finally computed the percentage of the logo that is shaded. The step-by-step approach, along with the verification of our answer, ensures accuracy and reinforces our understanding of the geometrical principles involved. We also addressed common mistakes and highlighted the practical applications of percentage calculations in geometry across various fields. This exercise not only enhances our problem-solving skills but also demonstrates the relevance of mathematical concepts in real-world scenarios. Through careful calculation and a clear understanding of geometrical relationships, we arrived at the solution of 84%.