Question

find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. express arc length in terms of π. then round your answer to two decimal places.
radius, r = 19 feet; central angle, θ = 340°
s = \frac{323π}{9} feet
(simplify your answer. type an exact answer in terms of π. use integers or fractions for any numbers in the expression.)
s = □ feet
(round to two decimal places as needed)

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( s=\frac{\theta}{360^{\circ}}\times2\pi r \).
Given \( r = 19 \) feet and \( \theta=340^{\circ} \), substitute these values into the formula:
\( s=\frac{340^{\circ}}{360^{\circ}}\times2\pi\times19 \)

Step2: Simplify the expression

First, simplify \( \frac{340}{360}=\frac{17}{18} \), then:
\( s=\frac{17}{18}\times38\pi=\frac{17\times38\pi}{18}=\frac{646\pi}{18}=\frac{323\pi}{9} \) (this is the exact form in terms of \( \pi \), which matches the given expression).

Step3: Calculate the decimal approximation

Now, to find the decimal value, we know that \( \pi\approx3.14159 \), so:
\( s=\frac{323\times3.14159}{9} \)
First, calculate \( 323\times3.14159 = 323\times3 + 323\times0.14159=969+45.73357 = 1014.73357 \)
Then, divide by 9: \( \frac{1014.73357}{9}\approx112.75 \) (rounded to two decimal places)

Answer:

The exact arc length is \( \frac{323\pi}{9} \) feet, and the approximate length (rounded to two decimal places) is \( 112.75 \) feet. For the boxed decimal answer: \( \boxed{112.75} \)