Question

what is the difference of the lengths of (widehat{bd}) and (widehat{ce})? use the value (pi = 3.14), and round the answer to two decimal places. image of two concentric circles with center a, angle (45^circ), (ab = 6) units, (bc = 2) units, points b, c on the horizontal radius, d on the inner circle, e on the outer circle
a. 1.25 units
b. 1.57 units
c. 2.56 units
d. 2.84 units

Explanation:

Step1: Find the radius of each circle

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

Step2: Recall the arc length formula

The formula for the length of an arc 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, \(\theta = 45^\circ\).

Step3: Calculate the length of \(\widehat{BD}\)

For arc \(\widehat{BD}\), \(r = 6\) and \(\theta=45^\circ\).
\[

$$\begin{align*} L_{BD}&=\frac{45^\circ}{360^\circ}\times2\times3.14\times6\\ &=\frac{1}{8}\times37.68\\ & = 4.71 \end{align*}$$

\]

Step4: Calculate the length of \(\widehat{CE}\)

For arc \(\widehat{CE}\), \(r = 8\) and \(\theta = 45^\circ\).
\[

$$\begin{align*} L_{CE}&=\frac{45^\circ}{360^\circ}\times2\times3.14\times8\\ &=\frac{1}{8}\times50.24\\ &=6.28 \end{align*}$$

\]

Step5: Find the difference between the two arc lengths

\[

$$\begin{align*} \text{Difference}&=L_{CE}-L_{BD}\\ &=6.28 - 4.71\\ &=1.57 \end{align*}$$

\]

Answer:

B. 1.57 units