Question

1. find the length of \\(\overarc{ab}\\). write your answer in exact or rounded form.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( \widehat{AB} \) is \( L=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.
Here, \( \theta = 105^\circ \) and \( r = 8 \) units.

Step2: Substitute values into formula

Substitute \( \theta = 105^\circ \) and \( r = 8 \) into the formula:
\( L=\frac{105^\circ}{360^\circ}\times2\pi\times8 \)

Step3: Simplify the expression

First, simplify \( \frac{105}{360}=\frac{7}{24} \), and \( 2\times8 = 16 \).
So, \( L=\frac{7}{24}\times16\pi \).
Simplify \( \frac{7\times16}{24}\pi=\frac{112}{24}\pi=\frac{14}{3}\pi \approx 14.66 \) (if we calculate the decimal approximation, \( \pi\approx3.1416 \), \( \frac{14}{3}\times3.1416\approx14.66 \)).

Answer:

The exact length of \( \widehat{AB} \) is \( \frac{14}{3}\pi \) units (or approximately \( 14.66 \) units).