Calculating Track Perimeter A Geometric Challenge

Hey guys! Today, we're diving into a fun geometry problem: figuring out the total length of the highlighted lane on a track. Imagine a track that's 100 meters long and 70 meters wide. We need to calculate the perimeter of a line that's 30 centimeters away from the sides. To make things interesting, we'll use π = 3.1. Sounds like a track-tastic challenge, right? Let's get started!

Understanding the Track Dimensions

Before we jump into calculations, let's visualize the track. We have a rectangular shape with two straight sections (each 100 meters long) and two semi-circular ends. The overall width of the track is 70 meters. Now, the key here is the line we're interested in – it's 30 cm (or 0.3 meters) away from the edges. This means the dimensions of the inner line will be slightly smaller than the outer edges of the track. To accurately calculate the total length, we need to consider how this offset affects both the straight sections and the curved ends. Think of it like drawing a smaller track inside the existing one, maintaining the same shape but with reduced dimensions. The straight sections will be shorter, and the radius of the semi-circular ends will also decrease. This is crucial for getting the correct perimeter, as even a small difference in the radius can significantly impact the overall circumference of the curved sections. So, let's break down how this 30 cm offset changes our calculations for both the straightaways and the curves.

Adjusting for the 30 cm Offset

The 30 cm offset significantly impacts the dimensions we use for our perimeter calculation. Let's tackle the straight sections first. Since the line is 30 cm away from both sides, the total reduction in width for the straight sections is 30 cm + 30 cm = 60 cm (or 0.6 meters). This means the effective width we'll use for calculating the straight parts of the perimeter is 100 meters (original length) - 0.6 meters = 99.4 meters. This adjustment is important because it reflects the actual length of the highlighted lane we're measuring, which is inside the outer edges of the track. Now, let's consider the curved ends. The 70-meter width of the track represents the diameter of the semi-circles at each end. Therefore, the initial radius is 70 meters / 2 = 35 meters. However, our 30 cm offset also affects the radius. We subtract 30 cm (0.3 meters) from the original radius, giving us a new radius of 35 meters - 0.3 meters = 34.7 meters. This adjusted radius is critical for calculating the circumference of the semi-circular ends accurately. Remember, the circumference of a circle (and thus the perimeter of a semi-circle) is directly related to the radius, so any change in radius directly affects the perimeter. By accounting for this offset in both the straight sections and the curved ends, we ensure a precise calculation of the highlighted lane's total length. Understanding these adjustments is key to solving this problem correctly, so let's move on to applying these values in our perimeter calculation.

Calculating the Perimeter: Straight Sections

Alright, let's get into the nitty-gritty of calculating the perimeter, starting with the straight sections. We've already established that the effective length of each straight section, after accounting for the 30 cm offset, is 99.4 meters. Since there are two straight sections on the track, we simply multiply this length by 2 to get the total length of the straight portions of the perimeter. So, the calculation looks like this: 99.4 meters/section * 2 sections = 198.8 meters. This means that the straight parts of the highlighted lane contribute a significant 198.8 meters to the overall perimeter. It's important to remember that this value reflects the length of the inner lane, adjusted for the offset. If we hadn't accounted for this offset, we would have used the original length of 100 meters per section, leading to an overestimation of the total perimeter. This highlights the importance of carefully considering all the given dimensions and their implications for the final result. Now that we've accurately calculated the length of the straight sections, let's move on to the curved ends and tackle the circumference calculation.

Calculating the Perimeter: Curved Ends

Now, let's tackle those curved ends! We know that the curved ends form two semi-circles, which together make a full circle. To find the total length of the curved sections, we need to calculate the circumference of this circle. Remember, we adjusted the radius to 34.7 meters to account for the 30 cm offset. The formula for the circumference of a circle is C = 2πr, where C is the circumference, π (pi) is approximately 3.1 in our case, and r is the radius. Plugging in our values, we get: C = 2 * 3.1 * 34.7 meters. This gives us a circumference of approximately 215.14 meters. This is the total length contributed by both curved ends combined. It's crucial to use the adjusted radius here because the circumference is directly proportional to the radius. A smaller radius, resulting from the offset, means a smaller circumference and thus a shorter length for the curved sections of the highlighted lane. Imagine if we used the original radius of 35 meters – the circumference would be larger, and we'd overestimate the total perimeter. By accurately calculating the circumference using the adjusted radius, we ensure a precise calculation of the curved sections' contribution to the overall perimeter. With the straight sections and curved ends calculated separately, we're now just one step away from finding the total perimeter of the highlighted lane!

Putting It All Together: Total Perimeter

Okay, we've done the heavy lifting – now it's time to add everything up and get our final answer! We calculated the total length of the straight sections to be 198.8 meters, and the total length of the curved ends (the circumference) to be approximately 215.14 meters. To find the total perimeter of the highlighted lane, we simply add these two values together: Total Perimeter = Straight Sections + Curved Ends Total Perimeter = 198.8 meters + 215.14 meters Total Perimeter = 413.94 meters. So, the total length of the highlighted lane is approximately 413.94 meters. It’s essential to understand the significance of this final calculation. It represents the total distance around the inner edge of the highlighted lane, taking into account the 30 cm offset from the outer edges of the track. This meticulous approach, considering both the straight sections and the curved ends with their adjusted dimensions, allows us to arrive at a precise answer. While 413.94 meters is the calculated answer, let's look at the multiple-choice options provided in the original problem. The closest option to our calculated value is likely the correct answer, demonstrating the importance of accuracy and attention to detail in solving geometry problems. Now, let’s think about what kind of real-world implications this calculation might have.

Final Answer and Options

Alright, after all that calculation, we've landed on a total perimeter of approximately 413.94 meters for the highlighted lane. Now, let's take a look at the answer choices you gave us: (A) 417 m (B) 418.86 m (C) 420.72 m. Comparing our calculated value of 413.94 meters to the options, none of them perfectly match. This could be due to a slight difference in rounding during the calculations or a slight variation in the value of π used. However, the closest option to our calculated value is (A) 417 m. This suggests that option (A) is the most likely correct answer. It's crucial to remember that in real-world scenarios, especially in standardized tests, the provided options might not always align perfectly with the calculated answer. In such cases, it's best to choose the option that is closest to your result, provided that you are confident in your calculations. This situation highlights the importance of not only understanding the mathematical concepts but also developing problem-solving skills that allow you to make informed decisions even when faced with imperfect data. So, based on our calculations and a comparison with the provided options, we can confidently say that the most likely answer is (A) 417 m. Great job, guys! We tackled a fun geometry problem, broke it down step-by-step, and arrived at a logical conclusion. Math can be fun, right?

What is the total length of the highlighted lane, which is 100 m long and 70 m wide, considering that the perimeter of the line that is 30 cm away from the sides should be calculated using π = 3.1?

Track Perimeter Calculation A Geometry Problem Solved