Liliana's Vase Base Problem Solving For Circular Area
Liliana, an aspiring ceramic artist, is embarking on a new project: crafting a beautiful vase with a circular base. She envisions a base that is neither too small nor too large, aiming for an area that falls within a specific range. This range is crucial for the vase's stability and aesthetic appeal. Liliana wants the area of the base to be between 135 square centimeters ($cm^2$) and 155 square centimeters ($cm^2$). To achieve this, she needs to determine the appropriate radius for the circular base. Our task is to help Liliana identify which circle could represent the ideal base for her vase, using 3.14 as an approximation for $\pi$. This involves understanding the relationship between a circle's radius and its area, and then applying this knowledge to find a radius that results in an area within Liliana's desired range.
Understanding the Area of a Circle
The area of a circle is a fundamental concept in geometry. It represents the amount of space enclosed within the circle's boundary. The formula for calculating the area of a circle is given by:
Where:
- A represents the area of the circle.
-
\pi$ (pi) is a mathematical constant approximately equal to 3.14159, often rounded to 3.14 for practical calculations.
- r represents the radius of the circle, which is the distance from the center of the circle to any point on its circumference.
This formula highlights the direct relationship between the radius and the area of a circle. As the radius increases, the area increases exponentially. This understanding is crucial for Liliana as she determines the appropriate radius for her vase's base.
Applying the Area Formula to Liliana's Project
In Liliana's case, we know the desired area range (135 $cm^2$ to 155 $cm^2$) and the approximate value of $\pi$ (3.14). Our goal is to find the radius r that corresponds to an area within this range. To do this, we can rearrange the area formula to solve for r:
This rearranged formula allows us to calculate the radius if we know the area. We can use this to find the minimum and maximum radii that Liliana needs for her vase base.
Determining the Minimum Radius
First, let's determine the minimum radius required to achieve an area of 135 $cm^2$. We'll plug the minimum area (135) and the value of $\pi$ (3.14) into our rearranged formula:
Calculating this gives us:
r_{min} = \sqrt{42.99} \approx 6.56$ cm Therefore, the minimum radius for the vase base is approximately 6.56 cm. Any circle with a radius smaller than this will have an area less than 135 $cm^2$, which is outside Liliana's desired range. ## Determining the Maximum Radius Next, let's determine the maximum radius allowed to keep the area within 155 $cm^2$. We'll use the same rearranged formula, but this time with the maximum area (155): $r_{max} = \sqrt{\frac{155}{3.14}}
Calculating this gives us:
r_{max} = \sqrt{49.36} \approx 7.03$ cm Therefore, the maximum radius for the vase base is approximately 7.03 cm. Any circle with a radius larger than this will have an area greater than 155 $cm^2$, again outside Liliana's desired range. ## Identifying the Suitable Circle Now that we have the minimum radius (6.56 cm) and the maximum radius (7.03 cm), we can identify which circle could represent the base of Liliana's vase. Any circle with a radius between 6.56 cm and 7.03 cm will have an area between 135 $cm^2$ and 155 $cm^2$, meeting Liliana's requirements. To make this more concrete, let's consider a few examples: * **Circle with a radius of 6.5 cm:** This radius is slightly smaller than our calculated minimum (6.56 cm), so its area would be less than 135 $cm^2$. This circle would not be suitable. * **Circle with a radius of 6.7 cm:** This radius falls within our desired range (6.56 cm to 7.03 cm). Its area would be between 135 $cm^2$ and 155 $cm^2$, making it a suitable option. * **Circle with a radius of 7.0 cm:** This radius is also within our desired range. Its area would be close to, but still less than, 155 $cm^2$, making it another suitable option. * **Circle with a radius of 7.1 cm:** This radius is slightly larger than our calculated maximum (7.03 cm), so its area would be greater than 155 $cm^2$. This circle would not be suitable. ## Conclusion: Helping Liliana Choose the Right Base In conclusion, by understanding the formula for the area of a circle and applying it to Liliana's specific requirements, we've determined that the radius of the vase's base must be between 6.56 cm and 7.03 cm. This allows Liliana to select a circle that will give her a base area between 135 $cm^2$ and 155 $cm^2$, ensuring the stability and aesthetic appeal of her vase. This exercise highlights the practical application of geometry in everyday situations, allowing artists like Liliana to bring their creative visions to life with mathematical precision. By calculating the minimum and maximum radii, we've provided Liliana with a clear guideline for choosing the perfect circular base for her vase. This not only ensures that the vase meets her desired size requirements but also demonstrates the importance of mathematical principles in artistic endeavors. The **area of a circle** plays a critical role in various applications, and Liliana's vase project serves as a tangible example of its significance. Through this process, Liliana has gained a deeper understanding of the relationship between a circle's radius and its area, empowering her to make informed decisions about her artistic creations. The use of the formula $A = \pi r^2$ and its rearranged form $r = \sqrt{\frac{A}{\pi}}$ has been instrumental in solving this practical problem. This knowledge will undoubtedly benefit her in future projects, allowing her to create even more stunning ceramic pieces. The **precision in measurement** is crucial in art and design, and Liliana's commitment to accuracy will contribute to the overall quality of her work. This problem-solving approach not only helps Liliana with her current project but also enhances her critical thinking skills. The ability to apply mathematical concepts to real-world scenarios is a valuable asset in any field. By successfully navigating this challenge, Liliana has demonstrated her aptitude for both artistic expression and mathematical reasoning. The **intersection of art and mathematics** is often overlooked, but this example showcases how the two disciplines can complement each other to produce exceptional results. Ultimately, Liliana's vase project underscores the importance of mathematical literacy in various aspects of life. Whether it's designing a ceramic vase or planning a garden, understanding geometric principles can lead to more successful and aesthetically pleasing outcomes. The **application of mathematical formulas** in practical situations empowers individuals to make informed decisions and achieve their goals. Liliana's journey in finding the perfect circular base for her vase is a testament to the power of mathematics in the creative process.
