Question

suppose the radius r is 3 units and the angle \\(\theta\\) is \\(\frac{\pi}{6}\\) radians. what is the sector area a? please round to two places after the decimal point.

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area \( A \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( A=\frac{1}{2}r^{2}\theta \).

Step2: Substitute the given values

We are given that \( r = 3 \) units and \( \theta=\frac{\pi}{6} \) radians. Substitute these values into the formula:
\( A=\frac{1}{2}\times(3)^{2}\times\frac{\pi}{6} \)
First, calculate \( (3)^{2}=9 \). Then the expression becomes \( A=\frac{1}{2}\times9\times\frac{\pi}{6} \).
Simplify \( \frac{1}{2}\times9=\frac{9}{2} \), so \( A = \frac{9}{2}\times\frac{\pi}{6}=\frac{9\pi}{12}=\frac{3\pi}{4} \).
Now, calculate the numerical value. \( \pi\approx3.14159 \), so \( \frac{3\times3.14159}{4}=\frac{9.42477}{4} = 2.3561925\).

Step3: Round to two decimal places

Rounding \( 2.3561925 \) to two decimal places, we look at the third decimal place which is 6. Since \( 6\gt5 \), we round up the second decimal place. So \( 2.3561925\approx2.36 \).

Answer:

\( 2.36 \)