find the length of the arc indicated in bold. leave your answer in terms of pi. radius 7 yards angle measure is 105

To find the length of an arc, you can use the formula:

\[ L = r \theta \]

where:

• \( L \) is the length of the arc,
• \( r \) is the radius of the circle,
• \( \theta \) is the angle in radians.

First, we need to convert the angle from degrees to radians. The conversion from degrees to radians is given by:

\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]

For your angle of 105 degrees:

\[ \theta_{\text{radians}} = 105 \times \frac{\pi}{180} = \frac{105\pi}{180} = \frac{7\pi}{12} \]

Now substituting the values into the arc length formula:

\[ L = r \theta = 7 \times \frac{7\pi}{12} = \frac{49\pi}{12} \]

Thus, the length of the arc is:

\[ \frac{49\pi}{12} \text{ yards} \]

Yes, let's double-check the calculations step-by-step for clarity.

1. Angle Conversion: The angle given is 105 degrees. To convert this angle to radians, we use the formula:

   \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]

   Substituting in the angle: \[ \theta_{\text{radians}} = 105 \times \frac{\pi}{180} = \frac{105\pi}{180} \]

   Simplifying this fraction: \[ \frac{105\pi}{180} = \frac{7\pi}{12} \]

2. Arc Length Calculation: The formula for the length of the arc is: \[ L = r \theta \]

   Here, the radius \( r = 7 \) yards and the angle \( \theta = \frac{7\pi}{12} \):

   \[ L = 7 \times \frac{7\pi}{12} = \frac{49\pi}{12} \]

Thus, the length of the arc is indeed:

\[ \frac{49\pi}{12} \text{ yards} \]

So, the calculations are accurate. The length of the arc is \( \frac{49\pi}{12} \) yards.