Calculate Arc Length: Radius 18.3 Cm, Angle 4π/3 Radians

Hey math whizzes and curious minds! Today, we're diving into a super cool geometry problem: figuring out the length of an arc intercepted by a central angle in a circle. You know, that curved little piece of the circle's edge that a slice of pizza (or a piece of pie!) would cover. We've got some specific numbers to work with: a radius (r) of 18.3 cm and a central angle ($ heta$) of 4π/3 radians. Our mission, should we choose to accept it (and we totally will!), is to find this arc length and round it to three significant digits. Let's get this mathematical party started!

Understanding the Arc Length Formula: The Key to Unlocking the Answer

Alright guys, so before we crunch any numbers, we need to get our heads around the formula for arc length. It's actually pretty straightforward once you see it. The length of an arc (s) is directly proportional to the radius (r) of the circle and the central angle ($ heta$) measured in radians. The magic formula that ties it all together is:

$$s = r \theta$$

See? Simple, right? Here's why this works, in a nutshell. A full circle has an angle of $2\pi$ radians, and its circumference (the total length around the circle) is $C = 2\pi r$. If you think about it, the arc length is just a fraction of the total circumference. The fraction is determined by the central angle relative to the full circle's angle ($2\pi$ radians). So, if the angle is $\theta$, the fraction is $\frac{\theta}{2\pi}$. Multiply this fraction by the circumference, and you get:

$$s = \left( \frac{\theta}{2\pi} \right) (2\pi r)$$

When you simplify that, the $2\pi$'s cancel out, and BAM! You're left with $s = r\theta$. Pretty neat, huh? This formula is your best friend when dealing with arc lengths, especially when the angle is already given in radians, which is exactly our situation here. So, make sure you've got this formula memorized – it's a fundamental concept in understanding circles and their parts.

Plugging in the Values: Let's Get Our Hands Dirty!

Now for the fun part – putting our given numbers into the formula! We are given:

• Radius (r): 18.3 cm
• Central Angle ($ heta$): $\frac{4 \pi}{3}$ radians

And our trusty formula is: $s = r\theta$.

So, we just substitute our values in:

$$s = (18.3 ext{ cm}) \times \left( \frac{4 \pi}{3} \right)$$

Before we whip out the calculator, let's simplify this a bit to make it easier. We can multiply the 18.3 by 4 and then divide by 3, or we can divide 18.3 by 3 first and then multiply by 4. Let's do the division first, as it often leads to cleaner numbers. $18.3$ divided by $3$ is $6.1$.

$$s = (6.1 ext{ cm}) \times (4 \pi)$$

Now, multiply $6.1$ by $4$: $6.1 \times 4 = 24.4$.

$$s = 24.4 \pi ext{ cm}$$

So, the exact arc length is $24.4 \pi$ cm. This is a perfectly valid answer if you're asked for the exact value. However, the problem asks us to round the length to three significant digits. This means we need to approximate the value of $\pi$ and perform the calculation.

Calculating the Numerical Value and Rounding to Three Significant Digits

Okay, time to bring out the calculator and get a numerical answer. We know that $\pi$ is approximately $3.14159265...$. We need to multiply $24.4$ by this value. Let's do that:

$$s \approx 24.4 \times 3.14159265... ext{ cm}$$

Using a calculator, $24.4 \times \pi$ gives us approximately $76.6990819...$ cm.

Now, the crucial step: rounding to three significant digits. What does that mean, you ask? Significant digits are the digits in a number that carry meaningful contribution to its measurement resolution. In simpler terms, they are the digits that are not zero, unless they are between two other significant digits. For our number, $76.6990819...$, let's identify the first three significant digits. These are the '7', the '6', and the next '6'.

So, we look at the first three digits: $76.6$. The next digit is a '9'. Because this digit ('9') is 5 or greater, we need to round up the last significant digit. In this case, the last significant digit is the second '6'. Rounding it up means it becomes a '7'.

Therefore, $76.6990819...$ rounded to three significant digits is $76.7$.

So, the length of the arc intercepted by the central angle $\theta = \frac{4 \pi}{3}$ radians in a circle with a radius of $r = 18.3$ cm is approximately 76.7 cm.

Why Radians Matter: A Quick Detour

It's super important to remember why we use radians in the arc length formula $s = r\theta$. If the angle were given in degrees, we'd have to convert it to radians first. Why? Because the radian measure inherently includes the factor of $2\pi$ for a full circle, making the formula $s = r\theta$ universally applicable when $\theta$ is in radians. If you used degrees, the formula would look different, involving a conversion factor: $s = \frac{\theta_{\text{degrees}}}{360^{\circ}} \times 2\pi r$. Using radians simplifies things immensely, and most higher-level math and physics problems will stick to radians. So, if you ever see an angle in degrees in an arc length problem, your first step should always be to convert it to radians. It's a small step that saves a lot of potential confusion and errors down the line. Always double-check the units of your angle – it's a common pitfall!

The Significance of Three Significant Digits: Precision in Measurement

Let's chat for a sec about three significant digits. When we talk about measurements in the real world, we can't be infinitely precise. The number of significant digits we report tells us about the precision of our measurement. For example, if we measured the radius as $18.3$ cm, those three digits imply a certain level of accuracy. Reporting our final answer to three significant digits ($76.7$ cm) ensures that our answer reflects the same level of precision as our input data. If we reported $76.700$ cm, it would imply a much higher precision than what our initial measurement of $18.3$ cm allows. On the other hand, reporting $77$ cm (two significant digits) would be losing some of the precision we actually have. So, rounding to three significant digits is a standard practice in science and engineering to maintain appropriate precision throughout calculations. It's about being accurate without overstating how precise your initial measurements were. Think of it as matching the output precision to the input precision – a fundamental concept in data handling and scientific reporting.

Final Thoughts: You've Mastered Arc Length!

And there you have it, folks! We've successfully calculated the length of an arc using the radius and a central angle in radians. We plugged our values into the $s = r\theta$ formula, performed the multiplication, and then carefully rounded our result to three significant digits. The final answer, 76.7 cm, is our precise measurement for that curved segment of the circle. Remember this process – it's a fundamental skill in geometry and trigonometry. Keep practicing, and soon you'll be calculating arc lengths like a pro! Happy calculating, everyone!