Work each problem. See Example 5. Arc Length A circular sector has an area of 50 in² . The radius of the circle is 5 in. What is the arc length of the sector?

Verified step by step guidance

1. Recall the formula for the area of a circular sector: \(\text{Area} = \frac{1}{2} r^2 \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
2. Substitute the given values into the area formula: \(50 = \frac{1}{2} \times 5^2 \times \theta\).
3. Simplify the equation to solve for \(\theta\): \(50 = \frac{1}{2} \times 25 \times \theta\) which simplifies to \(50 = 12.5 \times \theta\).
4. Isolate \(\theta\) by dividing both sides by 12.5: \(\theta = \frac{50}{12.5}\).
5. Use the arc length formula \(s = r \theta\) to find the arc length, substituting \(r = 5\) and the value of \(\theta\) found in the previous step.

Key Concepts

Area of a Circular Sector

The area of a circular sector is given by the formula A = (1/2) * r² * θ, where r is the radius and θ is the central angle in radians. This formula relates the sector's area to the radius and the angle, allowing calculation of one if the others are known.

Arc Length of a Circle

The arc length (s) of a sector is calculated by s = r * θ, where r is the radius and θ is the central angle in radians. This formula connects the linear distance along the circle's edge to the radius and angle.

Converting Between Area and Arc Length Using the Central Angle

To find the arc length from the sector's area and radius, first solve for the central angle θ using the area formula, then use θ to find the arc length. This process links the two measurements through the angle, enabling the solution.