Ex 6.1, 12 - Radius of an air bubble is increasing at 1/2 cm/s - Ex 6.1

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  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.1,12 The radius of an air bubble is increasing at the rate of 1﷮2﷯ cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm? Let r be the radius of air bubble Given 𝑑𝑟﷮𝑑𝑡﷯ = 1﷮2﷯ cm/s The Bubble is a sphere Volume of bubble = volume of sphere = 4﷮3﷯𝜋 𝑟﷮3﷯ we need to find the rate at which volume of the bubble is increasing when radius is 1 cm i.e. 𝑑𝑣﷮𝑑𝑡﷯ at 𝜋 = 1 cm 𝑑𝑣﷮𝑑𝑡﷯ = 𝑑 4﷮3﷯𝜋 𝑟﷮3﷯﷯﷮𝑑𝑡﷯ = 4﷮3﷯𝜋 𝑑 𝑟﷮3﷯﷮𝑑𝑡﷯ = 4﷮3﷯𝜋 𝑑 𝑟﷮3﷯﷮𝑑𝑟﷯ 𝑑𝑟﷮𝑑𝑡﷯ = 4𝜋﷮3﷯ 3𝑟﷮2﷯ 𝑑𝑟﷮𝑑𝑡﷯ = 4𝜋 𝑟﷮2﷯ 1﷮2﷯ = 2𝜋 𝑟﷮2﷯ = 2𝜋 When radius is 1cm 𝑑𝑣﷮𝑑𝑡﷯ = 2𝜋 1﷯﷮2﷯ 𝑑𝑣﷮𝑑𝑡﷯ = 2𝜋 Since volume is in cm3 & time is in sec So 𝑑𝑣﷮𝑑𝑡﷯ = 2𝜋 cm3/sec Hence, volume is increasing at rate of 2𝜋 cm3/sec when radius = 1cm

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