Question

find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. express the arc length in terms of π. then round your answer to two decimal places. radius, r = 11 inches; central angle, θ = 35° s = \frac{77π}{36} inches (simplify your answer. type an exact answer in terms of π. use integers or fractions for any numbers in the expression.) s = □ inches (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 \( s=\frac{\theta}{360^{\circ}}\times2\pi r \).

Step2: Substitute the given values

We are given \( r = 11 \) inches and \( \theta=35^{\circ} \). Substituting these values into the formula:
\( s=\frac{35^{\circ}}{360^{\circ}}\times2\pi\times11 \)
Simplify the expression:
First, simplify \( \frac{35}{360}\times2\times11 \). \( \frac{35}{360}\times2=\frac{35}{180}=\frac{7}{36} \), then \( \frac{7}{36}\times11=\frac{77}{36} \). So \( s = \frac{77\pi}{36} \) (this is the exact form).

Step3: Calculate the decimal approximation

To find the decimal value, we calculate \( \frac{77\pi}{36} \). We know that \( \pi\approx3.14159 \), so:
\( \frac{77\times3.14159}{36}=\frac{241.90243}{36}\approx6.72 \)

Answer:

The exact arc length is \( \frac{77\pi}{36} \) inches and the approximate arc length (rounded to two decimal places) is \( 6.72 \) inches.