Question

how long is an arc intercepted by the given central angle in a circle of radius 16.05 km? 60° the length of the intercepted arc is approximately □ km. (round to the nearest hundredth.)

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) intercepted by a central angle \( \theta \) (in degrees) in a circle of radius \( r \) is \( s=\frac{\theta}{360^{\circ}}\times2\pi r \). Here, \( \theta = 60^{\circ} \) and \( r = 16.05 \) km.

Step2: Substitute the values into the formula

First, calculate \( \frac{\theta}{360^{\circ}}=\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6} \). Then, \( 2\pi r=2\times\pi\times16.05 \). Now, multiply these two results: \( s = \frac{1}{6}\times2\times\pi\times16.05 \). Simplify this to \( s=\frac{16.05\times\pi}{3} \).

Step3: Calculate the numerical value

Using \( \pi\approx3.14159 \), we have \( s=\frac{16.05\times3.14159}{3} \). First, \( 16.05\times3.14159\approx50.4225 \). Then, divide by 3: \( \frac{50.4225}{3}\approx16.81 \).

Answer:

16.81