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Ex 9.4, 19 - Volume of spherical balloon being inflated changes - Variable separation - Statement given

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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Ex 9.4, 19 The volume of spherical balloon being inflated changes at a constant rate. if initially its radius is 3 units and after 3 seconds it is 6 units . Find the radius of balloon after 𝑡 seconds . Let volume of the spherical balloon = V V = 4﷮3﷯ 𝜋 𝑟﷮3﷯ Since volume changes at 𝑎 constant rate, ∴ 𝑑𝑉﷮𝑑𝑡﷯=𝑘 𝑑﷮𝑑𝑡﷯ 4﷮3﷯ 𝜋 𝑟﷮3﷯﷯ = k 4﷮3﷯ 𝜋 𝑑 𝑟﷮3﷯﷮𝑑𝑡﷯ = k 4﷮3﷯ 𝜋 3r2 𝑑𝑟﷮𝑑𝑡﷯ = k 4𝜋r2 𝑑𝑟﷮𝑑𝑡﷯ = k 4𝜋r2 𝑑𝑟 = k dt Integrating both sides 4𝜋 ﷮﷮r2 𝑑𝑟 = k ﷯ ﷮﷮𝑑𝑡﷯ 4𝜋 𝑟﷮3﷯﷮3﷯ = kt + C At T = 0, r = 3 units 4𝜋(3)﷮3﷯﷮3﷯ = k(0) + C 4𝜋(3)﷮2﷯ = C 4𝜋(9) = C 36𝜋 = C C = 36π Also, At T = 3, r = 6 units Putting t = 3, r = 6 and C = 36π in equation (1) 4𝜋(6)﷮3﷯﷮3﷯ = 3k + 36𝜋 4𝜋(216)﷮3﷯ = 3k + 36𝜋 288𝜋 = 3k + 36𝜋 252𝜋 = 3k 84𝜋 = k k = 84π Putting value of k & C in equation (1) 4𝜋𝑟﷮3﷯﷮3﷯ = 84𝜋𝑡 + 36𝜋 4𝜋𝑟﷮3﷯ = 3 84𝜋𝑡+36𝜋﷯ 4𝜋𝑟﷮3﷯ = 252𝜋t + 108𝜋 𝑟﷮3﷯ = 252𝜋𝑡+ 108𝜋 ﷮4𝜋﷯ 𝑟﷮3﷯ = 63t + 27 r = 63t + 27﷯﷮ 1﷮3﷯﷯ ∴ Radius of the balloon after 3 seconds is 63t + 27﷯﷮ 𝟏﷮𝟑﷯﷯ units

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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