Question

find the length s and area a.

(there is a circle with a sector, radius 2 ft, central angle \\(\frac{\pi}{3}\\), arc length s, and sector area a.)

\\(s = \square\\) feet
( type an integer or decimal rounded to three decimal places as needed. )

Explanation:

Step1: Calculate arc length s

The formula for arc length is $s = r\theta$, where $r=2$ ft, $\theta=\frac{\pi}{3}$.
$s = 2 \times \frac{\pi}{3}$

Step2: Compute numerical value for s

Calculate the decimal approximation:
$s = \frac{2\pi}{3} \approx 2.094$

Step3: Calculate sector area A

The formula for sector area is $A = \frac{1}{2}r^2\theta$.
$A = \frac{1}{2} \times 2^2 \times \frac{\pi}{3}$

Step4: Simplify and compute A

Simplify the expression and find the decimal:
$A = \frac{1}{2} \times 4 \times \frac{\pi}{3} = \frac{2\pi}{3} \approx 2.094$

Answer:

$s = 2.094$ feet
$A = 2.094$ square feet