Final answer:
The central angle of the arc with a length of 21 in a circle with a circumference of 252 units is 30 degrees, determined by setting up a proportion between the arc length and the circumference, corresponding to the angle and 360 degrees, respectively.
Step-by-step explanation:
To find the central angle of the arc in degrees, we use the fact that the whole circle consists of 360 degrees and the relationship between the arc length, the circumference of the circle, and the central angle of the arc.
The circumference (C) of the circle is given as 252 units, and the arc length (L) of the arc in question is 21 units. By setting up a proportion, we can find out how many degrees correspond to the arc length of 21, given that 360 degrees correspond to the full circumference of 252 units.
The proportion would be set up as follows:
C : L = 360° : central angle
252 : 21 = 360° : central angle
To solve for the central angle, we cross-multiply and divide:
21 × 360° = 252 × central angle
7560° = 252 × central angle
Central angle = 7560° / 252
When we carry out the division, we obtain:
Central angle = 30°
Therefore, the central angle of the arc with the length of 21 in a circle with a circumference of 252 units is 30 degrees.