Question

a sector of a circle has an area of 30 square feet. find the central angle which forms the sector if the radius is 6 feet.
the central angle of the sector is $\frac{5}{3}$ radians.
(type an integer or a simplified fraction.)
my score: 8/18 pts (44.44%)

Explanation:

Step1: Recall sector area formula

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

Step2: Plug in known values

Substitute \(A=30\), \(r=6\) into the formula:
\(30 = \frac{1}{2} \times 6^2 \times \theta\)

Step3: Simplify the equation

Calculate \(6^2=36\), so \(\frac{1}{2} \times 36 = 18\):
\(30 = 18\theta\)

Step4: Solve for \(\theta\)

Rearrange to isolate \(\theta\):
\(\theta = \frac{30}{18} = \frac{5}{3}\)

Answer:

$\frac{5}{3}$