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Question
The radius of an air bubble is increasing at the rate `1/2` cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
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Solution
Let the radius of the bubble = r and the volume of the bubble
`"V" = 4/3 pi"r"^3`
According to the question, `(dr)/dt = 1/2` cm/s
Again `(dV)/dt = d/dt (4/3 pir^3)`
`= d/dt (4/3 pir^3) * (dr)/dt `
Or `(dV)/dt 4pir^2 * 1/2 = 2 pir^2` cm3/s
`therefore ((dV)/dt)_(r = 1) = 2pi (1)^2`
`= 2 pi` cm3/s
Hence, the rate of increase of the volume of the bubble is 2π cm3/s.
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