Final answer:
Using related rates calculus, we find the rate at which the volume of a spherical bubble increases as 6 cm³/s when its surface area is increasing at a rate of 2 cm²/s and the radius is 6 cm. Option C is correct.
Step-by-step explanation:
The question is asking for the rate at which the volume of a spherical bubble is increasing when its surface area is increasing at a rate of 2 cm²/s and the radius of the bubble is 6 cm. To find the rate of increase in volume, we can use related rates calculus involving the formulas for the surface area and volume of a sphere.
The formula for the surface area (SA) of a sphere is SA = 4πr². The formula for the volume (V) of a sphere is V = ⅔3πr³. Given that d(SA)/dt = 2 cm²/s, and the radius (r) at which we need to find d(V)/dt is 6 cm, we first find the rate of change of the radius using:
d(SA)/dt = 8πr * d(r)/dt
2 = 8π*6 * d(r)/dt
d(r)/dt = 2/(48π) cm/s
Then we find the rate of increase in volume:
d(V)/dt = 4πr² * d(r)/dt
d(V)/dt = 4π*6² * 2/(48π)
d(V)/dt = 6 cm³/s, which corresponds to option C.