Determine the quadrant in which the terminal point of the arc having initial point at (1, 0) and having an arc length (1032/49)pi.

To determine the quadrant of the terminal point of the arc, we first need to find the angle of the arc in radians.

1. The arc length of a circle is given by the formula s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. In this case, we know that s = (1032/49)π and r = 1 (since the initial point is (1,0) which is one unit away from the origin), so we can solve for θ:

   θ = s/r = (1032/49)π / 1 = (1032/49)π radians

2. To find the quadrant of the terminal point, we need to know the angle in degrees. We can convert from radians to degrees using the conversion factor 180/π:

   θ = (1032/49)π * (180/π) = 1032/49 * 180 = 378.7755102 degrees

3. Now we can determine the quadrant. In a standard coordinate system:

   • Quadrant I is from 0 to 90 degrees
   • Quadrant II is from 90 to 180 degrees
   • Quadrant III is from 180 to 270 degrees
   • Quadrant IV is from 270 to 360 degrees

   Since 378.7755102 degrees is more than 360 degrees, we subtract 360 degrees to find the equivalent angle within the first revolution:

   θ' = 378.7755102 - 360 = 18.7755102 degrees

4. Therefore, the terminal point of the arc is in Quadrant I.