Questions: A circular arc of length 16 feet subtends a central angle of 75 degrees. Find the radius of the circle in feet. (Note: You can enter pi as 'pi' in your answer.)

Solution Steps

To find the radius of the circle given the arc length and the central angle, we can use the formula for the arc length of a circle: \( L = r \theta \), where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians. First, convert the angle from degrees to radians, then rearrange the formula to solve for the radius \( r \).

The central angle \( \theta \) is given in degrees as \( 75^\circ \). To use it in the arc length formula, we convert it to radians using the conversion factor \( \frac{\pi}{180} \): \[ \theta = 75 \times \frac{\pi}{180} = 1.3089969389957472 \text{ radians} \]

The formula for the arc length \( L \) of a circle is given by: \[ L = r \theta \] where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians. We can rearrange this formula to solve for the radius \( r \): \[ r = \frac{L}{\theta} \]

Substituting the known values into the formula: \[ r = \frac{16}{1.3089969389957472} \approx 12.2231 \]

Final Answer