Final answer:
The area of the sector AOB with an arc length of 6π cm and radius 9 cm is found using the formula for the area of a sector. The central angle θ is ½ radians, leading to an area of 20.25π cm².
Step-by-step explanation:
To calculate the area of a sector given the arc length (6π cm) and radius (9 cm), we use the formula for the area of a sector of a circle which is A = ½ * r² * θ, where θ is the central angle in radians. Since the circumference of the whole circle is 2π * r, we can find the angle θ by comparing the arc length to the entire circumference. The arc length of a circle with radius 9 cm is 6π cm, thus the circumference is 2π*9 = 18π cm. Hence, the fraction of the circumference that is the arc AB equals the fraction of the circle that is the sector AOB, so we have the proportion 6π / 18π = θ / (2π).
By simplifying, we get θ = ½ radians. Now, we can compute the area of the sector using the formula: A = ½ * 9² * ½ = ½ * 81 * ½ = ½ * 40.5 = 20.25. Therefore, the area of the sector AOB is 20.25π cm².