Question

11
the length of an arc of a circle is \\(\frac{26}{9}\pi\\) centimeters and the measure of the corresponding central angle is \\(65^\circ\\). what is the length of the circle’s radius?

a. \\(4\\) cm

b. \\(16\\) cm

c. \\(2\\) cm

d. \\(8\\) cm

Explanation:

Step1: Recall arc length formula

The formula for arc length $s$ is $s = \frac{\theta}{360^\circ} \times 2\pi r$, where $\theta$ is the central angle, $r$ is the radius.

Step2: Substitute given values

Given $s = \frac{26}{9}\pi$, $\theta = 65^\circ$. Substitute into the formula:
$\frac{26}{9}\pi = \frac{65^\circ}{360^\circ} \times 2\pi r$

Step3: Simplify the right-hand side

Simplify $\frac{65}{360} \times 2 = \frac{130}{360} = \frac{13}{36}$. The equation becomes:
$\frac{26}{9}\pi = \frac{13}{36}\pi r$

Step4: Solve for $r$

Divide both sides by $\pi$, then multiply both sides by $\frac{36}{13}$:
$r = \frac{26}{9} \times \frac{36}{13}$
Calculate $\frac{26}{13}=2$, $\frac{36}{9}=4$, so $r = 2 \times 4 = 8$

Answer:

D. 8 cm