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The surface area of a spherical bubble is increasing at the rate of 2cm²/s. When the radius of the bubble is 6 cm, then the rate by which the volume of the bubble increasing is.

A.6cm³/sec
B.9cm³/sec
C.3cm³/sec
D.13cm³/sec

1 Answer

2 votes

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.

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