Ex 6.1,9 - Chapter 6 Class 12 Application of Derivatives

Transcript

Ex 6.1,9 A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. Balloon is spherical let r be the radius of balloon . & v be the volume of balloon. We need to find. rate at which balloon volume is increasing with the radius when r is 10cm i.e. we need to find change of volume w.r.t radius when r = 10 i.e. we need to find 𝑑𝑣﷮𝑑𝑟﷯ when r = 10 we know that volume of sphere = 4﷮3﷯ πr3 v = 4﷮3﷯ πr3 Differentiate w. r. t radius 𝑑𝑣﷮𝑑𝑟﷯ = 𝑑 4﷮3﷯ 𝜋𝑟3﷯﷮𝑑𝑟﷯ 𝑑𝑣﷮𝑑𝑟﷯ = 4﷮3﷯ π 𝑑 𝑟3﷯﷮𝑑𝑟﷯ 𝑑𝑣﷮𝑑𝑟﷯ = 4﷮3﷯ π 3 𝑟﷮2﷯ 𝑑𝑣﷮𝑑𝑟﷯ = 4𝜋 𝑟﷮2﷯ When r = 10 𝑑𝑣﷮𝑑𝑟﷯ = 4 × π × (10)2 𝑑𝑣﷮𝑑𝑟﷯ = 400 π Since volume is in cm3 & Radius is in cm So, 𝑑𝑣﷮𝑑𝑟﷯ = 400π 𝒄𝒎𝟑﷮𝒄𝒎﷯ Hence, volume is increasing at the rate of 400 π cm3/cm when r = 10 cm