Calculating Arc Length Circle Y With Radius 3m And 70 Degree Central Angle

Hey guys! Today, we're diving into a fun geometry problem that involves calculating the arc length of a circle. Specifically, we're going to tackle a problem involving circle Y, which has a radius of 3 meters and a central angle XYZ measuring 70 degrees. Our mission is to find the approximate length of the minor arc XZ, rounding our answer to the nearest tenth of a meter. So, let's jump right in and break this down step by step!

Understanding Arc Length

Before we get into the nitty-gritty of the calculations, it’s crucial to understand what arc length actually means. Imagine you have a pizza, and you slice out a piece. The crust of that slice represents an arc, and the length of that crust is the arc length. In mathematical terms, the arc length is the distance along the curved line of the arc. It's a portion of the circumference of the circle. To calculate it, we need to consider two key elements the radius of the circle and the central angle that subtends the arc. The central angle is the angle formed at the center of the circle by the two radii that connect to the endpoints of the arc. In our case, the central angle XYZ is 70 degrees, which is the angle that "cuts out" the minor arc XZ from circle Y. Understanding this relationship between the radius, central angle, and arc length is fundamental to solving this problem accurately. Think of it as slicing a cake; the bigger the slice (angle), the longer the crust (arc length) you get. This intuitive understanding will help us in applying the formula and interpreting our results later on. So, keep this pizza analogy in mind as we move forward with the calculations. By grasping the concept of arc length clearly, we can avoid common pitfalls and arrive at the correct solution with confidence. Remember, geometry is all about visualizing shapes and understanding their properties, so take a moment to picture the arc within the circle, and you'll find the calculations much easier to follow. With a solid grasp of what we're trying to find, we're now ready to explore the formula that will help us calculate the arc length.

The Arc Length Formula

Now that we have a good handle on what arc length is, let's talk about the formula we'll use to calculate it. The arc length formula is derived from the relationship between the central angle, the radius of the circle, and the circumference of the circle. The formula is as follows:

Arc Length = (Central Angle / 360°) × 2πr

Where:

• Arc Length is what we're trying to find.
• Central Angle is the angle in degrees subtended by the arc at the center of the circle (in our case, 70°).
• π (pi) is a mathematical constant approximately equal to 3.14159.
• r is the radius of the circle (in our case, 3 meters).

This formula might seem a bit intimidating at first, but let's break it down to make it easier to understand. The fraction (Central Angle / 360°) represents the proportion of the circle's circumference that the arc covers. Think of it as a slice of a pie; if the central angle is 90°, then the arc covers 90/360 = 1/4 of the circle. The term 2πr represents the entire circumference of the circle. Remember, the circumference is the distance around the circle, and it's calculated by multiplying 2 by pi and the radius. So, when we multiply the fraction of the circle by the circumference, we get the length of the arc that corresponds to that fraction. This formula is a cornerstone in solving arc length problems, and understanding its components is crucial. The central angle dictates how much of the circle's circumference we're interested in, and the radius scales the circumference to the correct size. With this formula in our toolkit, we are well-equipped to tackle the specific problem at hand. Before we plug in the values, it's a good idea to mentally prepare ourselves for the units. Since the radius is given in meters, our arc length will also be in meters. This kind of unit awareness is a valuable practice in problem-solving, as it helps us catch any potential errors in our calculations. Now, let's move on to the next step and plug in the values from our problem into this powerful formula.

Plugging in the Values

Alright, now for the fun part! Let's take the values given in the problem and plug them into our arc length formula. Remember, we have:

• Central Angle = 70°
• Radius (r) = 3 meters

So, our formula looks like this:

Arc Length = (70° / 360°) × 2π(3)

The first thing we need to do is simplify the fraction 70° / 360°. Both 70 and 360 are divisible by 10, so we can simplify this to 7 / 36. Now, let's rewrite the formula with this simplified fraction:

Arc Length = (7 / 36) × 2π(3)

Next, let's deal with the 2π(3) part. We can multiply 2 and 3 to get 6, so we have:

Arc Length = (7 / 36) × 6π

Now, we can further simplify by dividing 6 by 36. 6 goes into 36 six times, so we get:

Arc Length = (7 / 6) × π

This is where we are going to use the approximate value of π, which is roughly 3.14159. So, our equation becomes:

Arc Length = (7 / 6) × 3.14159

Plugging in the values correctly is a critical step, and it's always a good idea to double-check your work. Make sure you've used the right numbers for the central angle and the radius, and that you've placed them in the correct spots in the formula. A small mistake here can lead to a completely different answer, so attention to detail is key. Simplifying the equation step by step, as we've done, makes the calculation process more manageable and less prone to errors. By breaking down the formula into smaller parts, we can focus on each part individually and ensure that we're handling it correctly. This methodical approach is a valuable skill in mathematics and problem-solving in general. Now that we've successfully plugged in the values and simplified the equation, we're ready to perform the final calculation and find the approximate arc length. Let's move on to the next section where we'll crunch the numbers and get our answer.

Calculating the Result

Okay, we've reached the final stretch! We've got our simplified equation:

Arc Length = (7 / 6) × 3.14159

Now, let's do the math. First, we'll multiply 7 by 3.14159:

7 × 3.14159 ≈ 21.99113

So, our equation now looks like this:

Arc Length ≈ 21.99113 / 6

Next, we'll divide 21.99113 by 6:

21.99113 / 6 ≈ 3.665188

We've calculated the arc length to be approximately 3.665188 meters. However, the problem asks us to round to the nearest tenth of a meter. So, we look at the digit in the hundredths place, which is 6. Since 6 is greater than or equal to 5, we round up the digit in the tenths place.

Therefore, the approximate length of minor arc XZ, rounded to the nearest tenth of a meter, is 3.7 meters.

When performing calculations, especially with decimals, it’s essential to be precise and carry enough decimal places throughout the process. Rounding too early can lead to inaccuracies in the final answer. In this case, we kept several decimal places until the very end, ensuring that our rounded result is as accurate as possible. The process of rounding also requires careful attention. Understanding the rules of rounding and applying them correctly is crucial for obtaining the correct final answer. Remember, if the digit to the right of the place you're rounding to is 5 or greater, you round up; otherwise, you round down. In our case, the digit in the hundredths place (6) was greater than 5, so we rounded up the digit in the tenths place. Now that we've successfully calculated the arc length and rounded it to the nearest tenth of a meter, we have our answer! We're almost there, just one more step to go. Let's move on to the final section where we'll choose the correct answer from the given options.

Selecting the Correct Answer

We've done the calculations, and we found that the approximate length of minor arc XZ is 3.7 meters. Now, let's look at the options provided in the problem:

A. 1.8 meters B. 3.7 meters C. 15.2 meters D. 18.8 meters

Comparing our calculated answer (3.7 meters) with the options, we can see that option B matches our result perfectly. Therefore, the correct answer is B. 3.7 meters.

Choosing the correct answer from the given options is the final step in the problem-solving process, and it's just as important as the calculations themselves. It's a good idea to double-check your work and make sure that your calculated answer aligns with one of the options. If your answer doesn't match any of the options, it's a sign that you might have made a mistake somewhere in your calculations, and you should go back and review your work. In this case, our calculated answer (3.7 meters) matched option B, which gives us confidence that we've solved the problem correctly. However, it's still a good practice to quickly review the steps we took to arrive at the answer, just to be absolutely sure. We started by understanding the concept of arc length and the arc length formula. Then, we plugged in the values from the problem into the formula, simplified the equation, and performed the calculations. Finally, we rounded our result to the nearest tenth of a meter and compared it with the given options. By following these steps carefully and methodically, we've successfully solved the problem and selected the correct answer. So, congratulations, guys! We've conquered another geometry challenge! Remember, practice makes perfect, so keep honing your skills and tackling new problems. With a solid understanding of the concepts and a systematic approach, you'll be able to solve any geometry problem that comes your way.

Conclusion

So, there you have it! We've successfully calculated the approximate length of minor arc XZ in circle Y. By understanding the concept of arc length, applying the arc length formula, plugging in the values correctly, and performing the calculations carefully, we found the answer to be 3.7 meters. This problem demonstrates the importance of understanding fundamental geometric principles and applying them systematically. Remember, geometry isn't just about memorizing formulas; it's about visualizing shapes, understanding their properties, and applying logical reasoning to solve problems. Keep practicing, keep exploring, and keep having fun with math! Whether you're slicing a pizza or designing a Ferris wheel, the principles of geometry are all around us, waiting to be discovered and applied. So, go forth and conquer those geometric challenges, guys! You've got this!