Questions: Find the length to three significant digits of the arc intercepted by a central angle θ in a circle of radius r. r=10.5 cm, θ=8π/5 radians The length of the intercepted arc is approximately cm. (Round to one decimal place as needed.)

Find the length to three significant digits of the arc intercepted by a central angle θ in a circle of radius r.
r=10.5 cm, θ=8π/5 radians

The length of the intercepted arc is approximately cm. (Round to one decimal place as needed.)

Transcript text: Find the length to three significant digits of the arc intercepted by a central angle $\theta$ in a circle of radius $r$. \[ \mathrm{r}=10.5 \mathrm{~cm}, \theta=\frac{8 \pi}{5} \text { radians } \] The length of the intercepted arc is approximately $\square$ cm. (Round to one decimal place as needed.)

Solution

Solution Steps

Step 1: Given Values

We are given the radius $ r = 10.5 \, \text{cm} $ and the central angle $ \theta = \frac{8\pi}{5} \, \text{radians} $.

Step 2: Calculate Arc Length

Using the formula for arc length $ s $: \[ s = r \theta \] we substitute the given values: \[ s = 10.5 \cdot \frac{8\pi}{5} \] Calculating this gives: \[ s \approx 52.778756580308524 \, \text{cm} \]

Step 3: Round to Significant Digits

Rounding the arc length to one decimal place, we find: \[ s \approx 52.8 \, \text{cm} \]