Question

The central angle in a circle of radius 6 meters has an intercepted arc length of 10 meters. Find the measure of the angle in radians and in degrees

Answer 1

To find the angle in radians, divide the arc length (10 meters) by the radius (6 meters), which equals 5/3 radians. To convert to degrees, multiply by 180/π, resulting in approximately 95.49 degrees.

Step-by-step explanation:

To calculate the measure of the central angle in radians when the radius of a circle is 6 meters and the intercepted arc length is 10 meters, you can use the formula for arc length, which is arc length (s) = radius (r) × angle (θ) in radians. Solving for the angle, we get θ = s / r. Substituting the given values, θ = 10 meters / 6 meters = 5/3 radians.

Converting this angle from radians to degrees, we multiply by the conversion factor (180°/π). Therefore, the angle in degrees is θ = (5/3) × (180°/π) ≈ 95.49°. The measure of the central angle is approximately 5/3 radians or 95.49 degrees.

Answer 2

The central angle is 5/3 radians or approximately 95.4930°.

Step-by-step explanation:

Recall that arc-length is given by the formula:

`\displaystyle s = r\theta`

Where s is the arc-length, r is the radius of the circle, and θ is the measure of the central angle, in radians.

Since the intercepted arc-length is 10 meters and the radius is 6 meters:

`\displaystyle (10) = (6)\theta`

Solve for θ:

`\displaystyle \theta = (5)/(3)\text{ rad}`

The central angle measures 5/3 radians.

Recall that to convert from radians to degrees, we can multiply by 180°/π. Hence:

`\displaystyle \frac{5\text{ rad}}{3} \cdot \frac{180^\circ}{\pi \text{ rad}} = (300)/(\pi)^\circ\approx 95.4930^\circ`

So, the central angle is approximately 95.4930°