Question

what is the difference of the lengths of \\(\overarc{bd}\\) and \\(\overarc{ce}\\)? use the value \\(\pi = 3.14\\), and round the answer to two decimal places.
\\(ab = 6\\) units
\\(bc = 2\\) units

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( \widehat{XY} \) with radius \( r \) and central angle \( \theta \) (in degrees) is \( L=\frac{\theta}{360}\times2\pi r \).

Step2: Find radius of \( \widehat{BD} \)

For arc \( \widehat{BD} \), the radius \( r_{BD}=AB = 6 \) units, and \( \theta = 45^\circ \).
Using the arc length formula: \( L_{BD}=\frac{45}{360}\times2\times3.14\times6 \)
Simplify: \( \frac{1}{8}\times37.68 = 4.71 \)

Step3: Find radius of \( \widehat{CE} \)

For arc \( \widehat{CE} \), the radius \( r_{CE}=AC=AB + BC=6 + 2=8 \) units, and \( \theta = 45^\circ \).
Using the arc length formula: \( L_{CE}=\frac{45}{360}\times2\times3.14\times8 \)
Simplify: \( \frac{1}{8}\times50.24 = 6.28 \)

Step4: Find the difference

Difference \( = L_{CE}-L_{BD}=6.28 - 4.71=1.57 \)

Answer:

\( 1.57 \)