Question

find the length of the arc, s, on a circle of radius r intercepted by a central angle \\(\theta\\). express arc length in terms of \\(\pi\\). then round your answer to two decimal places.\
radius, \\(r = 14\\) feet; central angle, \\(\theta = 325^\circ\\)\
\\(s = \square\\) feet\
(simplify your answer. type an exact answer in terms of \\(\pi\\). use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Convert degrees to radians

$\theta = 325^\circ \times \frac{\pi}{180^\circ} = \frac{65\pi}{36}$ radians

Step2: Apply arc length formula

$s = r\theta = 14 \times \frac{65\pi}{36}$

Step3: Simplify the exact expression

$s = \frac{14 \times 65\pi}{36} = \frac{910\pi}{36} = \frac{455\pi}{18}$

Step4: Calculate decimal approximation

$s \approx \frac{455 \times 3.1416}{18} \approx \frac{1429.428}{18} \approx 79.41$

Answer:

Exact value: $\frac{455\pi}{18}$ feet
Decimal approximation: $79.41$ feet