Ex 6.1,13 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.1, 13 A balloon, which always remains spherical, has a variable diameter 3/2 (2𝑥 +1). Find the rate of change of its volume with respect to 𝑥. Let d be the diameter of the balloon Given d = 3/2 (2x + 1) Let r be the radius of the balloon r = 𝑑/2 = 3/4 (2x + 1) The balloon is a sphere Volume of the balloon = Volume of sphere = 4/3 𝜋𝑟^3 We need to find rate of change of volume with respect to x that is 𝑑𝑣/𝑑𝑥 𝑑𝑣/𝑑𝑥 = 𝑑/𝑑𝑥 (4/3 𝜋𝑟^3 ) = 4𝜋/3 (𝑑𝑟^3)/𝑑𝑥 = 4𝜋/3 𝑑/𝑑𝑥 (27/64 (2𝑥+1)^3 ) = 9𝜋/16 (𝑑(2𝑥 + 1)^3)/𝑑𝑥 = 9𝜋/16 3(2x + 1)2 (2) = 𝟐𝟕𝝅/𝟖 (𝟐𝒙+𝟏)^𝟐 Hence, 𝑑𝑣/𝑑𝑥 = 27𝜋/8 (2𝑥+1)^2