Certainly! Let's work through this step-by-step.
### Step 1: Understand the Problem
We need to find the length of an arc in a circle with:
- Radius [tex]\( r = 7 \)[/tex] yards
- Central angle [tex]\( \theta = \frac{\pi}{8} \)[/tex] radians
### Step 2: Recall the Formula for Arc Length
The arc length [tex]\( s \)[/tex] in a circle is given by the formula:
[tex]\[ s = r \cdot \theta \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the central angle in radians.
### Step 3: Substitute the Given Values
By plugging in the given values [tex]\( r = 7 \)[/tex] yards and [tex]\( \theta = \frac{\pi}{8} \)[/tex] radians into the formula:
[tex]\[ s = 7 \cdot \frac{\pi}{8} \][/tex]
### Step 4: Calculate the Exact Arc Length
This results in the following calculation:
[tex]\[ s = 7 \cdot \frac{\pi}{8} \][/tex]
### Step 5: Simplify the Expression
We can express this more clearly as:
[tex]\[ s = \frac{7\pi}{8} \][/tex]
### Step 6: Calculate the Numerical Value
Let's approximate the numerical value of [tex]\( \frac{7\pi}{8} \)[/tex]:
[tex]\[ \frac{7\pi}{8} \approx 2.748893571891069 \][/tex]
### Step 7: Round the Result
Finally, round this value to two significant digits:
[tex]\[ s \approx 2.75 \][/tex]
### Conclusion
Therefore, the length of the arc is approximately 2.75 yards.
So, the answer is:
[tex]\[ s \approx 2.75 \text{ yards} \][/tex]
