To find the length of the arc \( \overset{\frown}{BD} \) in circle \( C \), we can use the formula for the length of an arc, which is given by:
\[ L = r \theta \]
where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians.
From the given information:
- We know that \( m \angle BCD = 66^\circ \).
- The length of the chord \( BC = 3 \) units.
To find the radius \( r \), we can relate the chord length \( BC \) to the radius and the angle \( \angle BCD \). The formula for the length of a chord in terms of the radius \( r \) and the angle \( \theta \) in radians is:
\[ BC = 2r \sin\left(\frac{\theta}{2}\right) \]
First, we convert the angle \( 66^\circ \) to radians:
\[ \theta = 66^\circ \times \frac{\pi}{180^\circ} = \frac{66\pi}{180} = \frac{11\pi}{30} \text{ radians} \]
Now we calculate the sine of half that angle:
\[ \frac{\theta}{2} = \frac{11\pi}{60} \]
Next, we set up the equation with \( BC = 3 \):
\[ 3 = 2r \sin\left(\frac{11\pi}{60}\right) \]
Now, solve for \( r \):
\[ r = \frac{3}{2 \sin\left(\frac{11\pi}{60}\right)} \]
Using a calculator to find \( \sin\left(\frac{11\pi}{60}\right) \):
\[ \sin\left(\frac{11\pi}{60}\right) \approx 0.5 \]
Then we can calculate \( r \):
\[ r \approx \frac{3}{2 \times 0.5} = \frac{3}{1} = 3 \]
Now that we have the radius \( r \approx 3 \), we can find the length of the arc \( \overset{\frown}{BD} \):
Using the full angle in radians:
\[ L = r \theta = 3 \times \frac{11\pi}{30} \]
Calculating this:
\[ L \approx 3 \times \frac{11 \times 3.14}{30} \approx 3 \times \frac{34.54}{30} \approx 3 \times 1.1513 \approx 3.454 \]
Rounding to the nearest hundredth:
\[ L \approx 3.45 \]
Thus, the length of the arc \( \overset{\frown}{BD} \) is approximately \( \boxed{3.45} \) units.