Certainly! Let's find the length of the arc made by a central angle [tex]\(\theta = 11^\circ\)[/tex] in a circle with a radius [tex]\(r = 2500\, \text{km}\)[/tex].
### Step 1: Convert Degrees to Radians
To calculate the arc length, we first need to convert the angle from degrees to radians. Remember that:
[tex]\[
1 \text{ degree} = \frac{\pi}{180} \text{ radians}
\][/tex]
So, we convert [tex]\(11^\circ\)[/tex] to radians:
[tex]\[
\theta \text{ (in radians)} = 11 \times \frac{\pi}{180}
\][/tex]
Using the value of [tex]\(\pi \approx 3.14159\)[/tex], we get:
[tex]\[
\theta \approx 11 \times 0.0174533 \approx 0.19198621771937624 \text{ radians}
\][/tex]
### Step 2: Use the Arc Length Formula
The length of the arc [tex]\(s\)[/tex] is given by the formula:
[tex]\[
s = r \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the angle in radians. Substituting the values we have:
[tex]\[
s = 2500 \times 0.19198621771937624 \approx 479.9655442984406 \text{ km}
\][/tex]
### Step 3: Round to Two Significant Digits
Finally, we round the arc length to two significant digits:
[tex]\[
s \approx 479.97 \text{ km}
\][/tex]
Thus, the approximate length of the arc is:
[tex]\[
s \approx 479.97 \text{ km}
\][/tex]
