Final answer:
To calculate the arc length s, we use the formula s = (\theta / 360) \times 2\pi r. With a diameter of 12 cm and central angle of 75 degrees, the radius is 6 cm, so s = 25/3\pi cm, which corresponds to option (b).
Step-by-step explanation:
To determine the length of the intercepted arc (s) in a sector of a circle with a central angle and a known diameter, one can use the relationship between the central angle and the circumference of the circle. The arc length formula is:
s = (\theta / 360) \times 2\pi r
where \theta is the central angle in degrees, r is the radius of the circle, and s is the arc length. In this case, the diameter of the circle is given as 12 cm, so the radius (r) is 6 cm (half of the diameter). The central angle (\theta) is 75 degrees. Applying these values to the formula gives:
s = (75 / 360) \times 2\pi \times 6
s = (75 / 360) \times 12\pi
s = (1 / 4.8) \times 12\pi
s = 2.5\pi cm
Thus, the correct length of the intercepted arc is 25/3\pi cm, which corresponds to option (b) 25/3 \pi cm.