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, angle at a is \\(45^\circ\\). options: a. 1.25 units, b. 1.57 units, c. 2.56 units, d. 2.84 units

Explanation:

Step1: Find radius of inner circle

The radius of the inner circle (for arc $\widehat{BD}$) is $AB = 6$ units.

Step2: Find radius of outer circle

The radius of the outer circle (for arc $\widehat{CE}$) is $AC = AB + BC = 6 + 2 = 8$ units.

Step3: Calculate length of $\widehat{BD}$

Use arc length formula $L = \frac{\theta}{360^\circ} \times 2\pi r$.
$\widehat{BD} = \frac{45^\circ}{360^\circ} \times 2 \times 3.14 \times 6 = \frac{1}{8} \times 37.68 = 4.71$ units

Step4: Calculate length of $\widehat{CE}$

$\widehat{CE} = \frac{45^\circ}{360^\circ} \times 2 \times 3.14 \times 8 = \frac{1}{8} \times 50.24 = 6.28$ units

Step5: Find the difference

Difference = $\widehat{CE} - \widehat{BD} = 6.28 - 4.71 = 1.57$ units

Answer:

B. 1.57 units