Questions: A sector of a circle has a diameter of 14 feet and an angle of (3 pi/5) radians. Find the area of the sector. Round your answer to four decimal places.

Solution Steps

To find the area of a sector of a circle, we can use the formula:

\[ \text{Area of sector} = \frac{\theta}{2\pi} \times \pi r^2 \]

where \(\theta\) is the angle in radians and \(r\) is the radius of the circle. Given the diameter, we can find the radius by dividing the diameter by 2. Then, substitute the values into the formula to calculate the area.

The diameter of the circle is given as 14 feet. The radius \( r \) is half of the diameter: \[ r = \frac{14}{2} = 7 \text{ feet} \]

The formula for the area of a sector is: \[ \text{Area of sector} = \frac{\theta}{2\pi} \times \pi r^2 \] where \( \theta = \frac{3\pi}{5} \) radians.

Substitute the values of \( \theta \) and \( r \) into the formula: \[ \text{Area of sector} = \frac{\frac{3\pi}{5}}{2\pi} \times \pi \times 7^2 \]

Simplify the expression: \[ \text{Area of sector} = \frac{3}{10} \times \pi \times 49 \] \[ \text{Area of sector} = \frac{147\pi}{10} \]

Calculate the numerical value of the area: \[ \text{Area of sector} \approx 46.1814 \text{ square feet} \]

Final Answer